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THEOBY OF FUNCTIONS 



OF A 



COMPLEX VARIABLE. 



Itonlion: C. J. CLAY AND SONS, 

CAMBEIDGE UNIVEESITY PEESS WAEEHOUSE, 

AVE MAEIA LANE. 




CAMBEIDGE : DEIGHTON, BELL, AKD CO. 

LEIPZIG : F. A. BROCKHAUS. 
NEW YORK: MACMILLAN AND CO. 



THEOEY OF FUNCTIONS 



OF A 



COMPLEX VARIABLE 



BY 



A. R FORSYTE, So.D., F.RS., 

FELLOW OF TRINITY COLLEGE, CAMBRIDGE. 



CAMBEIDGE: 
AT THE UNIVERSITY PRESS. 

1893 

All rights reserved. . 




Mtth. U. 01. 



PRINTED BY C. J. CLAY, M.A. AND SONS, 
AT THE UNIVERSITY PRESS. 



PEEFACE. 

AMONG the many advances in the progress of mathematical 
XlL science during the last forty years, not the least remarkable 
are those in the theory of functions. The contributions that are 
still being made to it testify to its vitality : all the evidence points 
to the continuance of its growth. And, indeed, this need cause no 
surprise. Few subjects can boast such varied processes, based 
upon methods so distinct from one another as are those originated 
by Cauchy, by Weierstrass, and by Biemann. Each of these 
methods is sufficient in itself to provide a complete development ; 
combined, they exhibit an unusual wealth of ideas and furnish 
unsurpassed resources in attacking new problems. 

It is difficult to keep pace with the rapid growth of the 
literature which is due to the activity of mathematicians, 
especially of continental mathematicians : and there is, in con 
sequence, sufficient reason for considering that some marshalling 
of the main results is at least desirable and is, perhaps, necessary. 
Not that there is any dearth of treatises in French and in 
German : but, for the most part, they either expound the pro 
cesses based upon some single method or they deal with the 
discussion of some particular branch of the theory. 



814033 



PREFACE 



The present treatise is an attempt to give a consecutive 
account of what may fairly be deemed the principal branches of 
the whole subject. It may be that the next few years will see 
additions as important as those of the last few years : this account 
would then be insufficient for its purpose, notwithstanding the 
breadth of range over which it may seem at present to extend. 
My hope is that the book, so far as it goes, may assist mathe 
maticians, by lessening the labour of acquiring a proper knowledge 
of the subject, and by indicating the main lines, on which recent 
progress has been achieved. 

No apology is offered for the size of the book. Indeed, if 
there were to be an apology, it would rather be on the ground 
of the too brief treatment of some portions and the omissions 
of others. The detail in the exposition of the elements of several 
important branches has prevented a completeness of treatment 
of those branches : but this fulness of initial explanations is 
deliberate, my opinion being that students will thereby become 
better qualified to read the great classical memoirs, by the study 
of which effective progress can best be made. And limitations of 
space have compelled me to exclude some branches which other 
wise would have found a place. Thus the theory of functions of 
a real variable is left undiscussed : happily, the treatises of Dini, 
Stolz, Tannery and Chrystal are sufficient to supply the omission. 
Again, the theory of functions of more than one complex variable 
receives only a passing mention ; but in this case, as in most 
cases, where the consideration is brief, references are given 
which will enable the student to follow the development to 
such extent as he may desire. Limitation in one other direction 
has been imposed : the treatise aims at dealing with the general 
theory of functions and it does not profess to deal with special 
classes of functions. I have not hesitated to use examples of 
special classes : but they are used merely as illustrations of the 
general theory, and references are given to other treatises for 
the detailed exposition of their properties. 



PREFACE Vll 

The general method which is adopted is not limited so that 
it may conform to any single one of the three principal inde 
pendent methods, due to Cauchy, to Weierstrass and to Biemann 
respectively : where it has been convenient to do so, I have 
combined ideas and processes derived from different methods. 

The book may be considered as composed of five parts. 

The first part, consisting of Chapters I VII, contains the 
theory of uniform functions : the discussion is based upon power- 
series, initially connected with Cauchy s theorems in integration, 
and the properties established are chiefly those which are con 
tained in the memoirs of Weierstrass and Mittag-Leffler. 

The second part, consisting of Chapters VIII XIII, contains 
the theory of multiform functions, and of uniform periodic 
functions which are derived through the inversion of integrals 
of algebraic functions. The method adopted in this part is 
Cauchy s, as used by Briot and Bouquet in their three memoirs 
and in their treatise on elliptic functions : it is the method that 
has been followed by Hermite and others to obtain the properties 
of various kinds of periodic functions. A chapter has been 
devoted to the proof of Weierstrass s results relating to functions 
that possess an addition-theorem. 

The third part, consisting of Chapters XIV XVIII, contains 
the development of the theory of functions according to the 
method initiated by Biemann in his memoirs. The proof which 
is given of the existence-theorem is substantially due to Schwarz ; 
in the rest of this part of the book, I have derived great assist 
ance from Neumann s treatise on Abelian functions, from Fricke s 
treatise on Klein s theory of modular functions, and from many 
memoirs by Klein. 

The fourth part, consisting of Chapters XIX and XX, treats 
of conformal representation. The fundamental theorem, as to the 
possibility of the conformal representation of surfaces upon one 
another, is derived from the existence-theorem : it is a curious fact 
that the actual solution, which has been proved to exist in general, 
F. b 



Vlll PREFACE 

has been obtained only for cases in which there is distinct 
limitation. 

The fifth part, consisting of Chapters XXI and XXII, contains 
an introduction to the theory of Fuchsian or automorphic functions, 
based upon the researches of Poincare and Klein : the discussion is 
restricted to the elements of this newly-developed theory. 

The arrangement of the subject-matter, as indicated in this 
abstract of the contents, has been adopted as being the most 
convenient for the continuous exposition of the theory. But the 
arrangement does not provide an order best adapted to one who is 
reading the subject for the first time. I have therefore ventured 
to prefix to the Table of Contents a selection of Chapters that 
will probably form a more suitable introduction to the subject for 
such a reader ; the remaining Chapters can then be taken in an 
order determined by the branch of the subject which he wishes 
to follow out. 

In the course of the preparation of this book, I have consulted 
many treatises and memoirs. References to them, both general 
and particular, are freely made : without making precise reserva 
tions as to independent contributions of my own, I wish in this 
place to make a comprehensive acknowledgement of my obligations 
to such works. A number of examples occur in the book : most of 
them are extracted from memoirs, which do not lie close to the 
direct line of development of the general theory but contain 
results that provide interesting special illustrations. My inten 
tion has been to give the author s name in every case where a 
result has been extracted from a memoir : any omission to do so 
is due to inadvertence. 

Substantial as has been the aid provided by the treatises and 
memoirs to which reference has just been made, the completion of 
the book in the correction of the proof-sheets has been rendered 
easier to me by the unstinted and untiring help rendered by 
two friends. To Mr William Burnside, M.A., formerly Fellow of 



PREFACE 



Pembroke College, Cambridge, and now Professor of Mathematics 
at the Royal Naval College, Greenwich, I am under a deep debt 
of gratitude : he has used his great knowledge of the subject in 
the most generous manner, making suggestions and criticisms that 
have enabled me to correct errors and to improve the book in 
many respects. Mr H. M. Taylor, M. A., Fellow of Trinity College, 
Cambridge, has read the proofs with great care : the kind assist 
ance that he has given me in this way has proved of substantial 
service and usefulness in correcting the sheets. I desire to 
recognise most gratefully my sense of the value of the work which 
these gentlemen have done. 

It is but just on my part to state that the willing and active 
co-operation of the Staff of the University Press during the pro 
gress of printing has done much to lighten my labour. 

It is, perhaps, too ambitious to hope that, on ground which 
is relatively new to English mathematics, there will be freedom 
from error or obscurity and that the mode of presentation in this 
treatise will command general approbation. In any case, my aim 
has been to produce a book that will assist mathematicians in 
acquiring a knowledge of the theory of functions : in proportion 
as it may prove of real service to them, will be my reward. 

A. R. FORSYTE. 



TRINITY COLLEGE, CAMBRIDGE. 
25 February, 1893. 



CONTENTS. 



The following course is recommended, in the order specified, to those who are 
reading the subject for the first time : The theory of uniform functions, Chapters 
I V ; Conformal representation, Chapter XIX ; Multiform functions and uniform 
periodic functions, Chapters VIII XI ; Riemanris surfaces, and Riemann s theory 
of algebraic functions and their integrals, Chapters XIV XVI, XVIII. 



CHAPTER I. 

GENERAL INTRODUCTION. 



PAGE 

13. The complex variable and the representation of its variation by points 

in a plane , 

4. Neumann s representation by points on a sphere ... 4 

5. Properties of functions assumed known ... Q 
6, 7. The idea of complex functionality adopted, with the conditions neces 
sary and sufficient to ensure functional dependence ... 6 

8. Riemann s definition of functionality ... g 

9. A functional relation between two complex variables establishes the 

geometrical property of conformal representation of their planes . 10 

10, 11. Relations between the real and the imaginary parts of a function of z 11 
12, 13. Definitions and illustrations of the terms monogenic, uniform, multiform, 

branch, branch-point, holomorphic, zero, pole, meromorphic . . . 14 



CHAPTER II. 

INTEGRATION OF UNIFORM FUNCTIONS. 

14, 15. Definition of an integral with complex variables ; inferences . . . 18 
16. Proof of the lemma I I (^ - \ dxdy=\(pdx -\-qdy), under assigned 

I \ fll 1 (it I I J * J. / f O 



conditions 21 



CONTENTS 



PAGE 

17, 18. The integral \f(z)dz round any simple curve is zero, when f(z) is 

Cz 

holomorphic within the curve; and I /(*)<& is a holomorphic 

J a 

function when the path of integration lies within the curve . . 23 
19. The path of integration of a holomorphic function can be deformed 

without changing the value of the integral ..... 26 

2022. The integral = . I - dz, round a curve enclosing a, is /(a) when 
27rt J z a 

f(z) is a holomorphic function within the curve; and the integral 

J_ [ /(*) dz is ,^. Superior limit for the modulus of 
27rt J(z-a) n + 1 n\ da n 

the nth derivative of /(a) in terms of the modulus of /(a) . . 27 

23. The path of integration of a meromorphic function cannot be deformed 

across a pole without changing the value of the integral. . . 34 

24. The integral of any function (i) round a very small circle, (ii) round a 

very large circle, (iii) round a circle which encloses all its infinities 

and all its branch-points ......... 35 

25. Special examples ............ 



CHAPTER III. 

EXPANSIONS OF FUNCTIONS IN SERIES OF POWERS. 

26, 27. Cauchy s expansion of a function in positive powers of z - a ; with re 
marks and inferences 43 

2830. Laurent s expansion of a function in positive and negative powers of 

z - a ; with corollary 47 

31. Application of Cauchy s expansion to the derivatives of a function . 51 

32, 33. Definition of an ordinary point of a function, of the domain of an 
ordinary point, of an accidental singularity, and of an essential 
singularity ....... 52 

34, 35. Continuation of a function by means of elements over its region of 

continuity 54 

36. Schwarz s theorem on symmetric continuation across the axis of real 

quantities 57 



CHAPTER IV. 

UNIFORM FUNCTIONS, PARTICULARLY THOSE WITHOUT ESSENTIAL 
SINGULARITIES. 

37. A function, constant over a continuous series of points, is constant 

everywhere in its region of continuity 59 

38, 39. The multiplicity of a zero, which is an ordinary point, is finite; and 

a multiple zero of a function is a zero of its first derivative . . 61 



CONTENTS Xlll 

PAGE 

40. A function, that is not a constant, must have infinite values . . 63 

41, 42. Form of a function near an accidental singularity 64 

43, 44. Poles of a function are poles of its derivatives ..... 66 
45, 46. A function, which has infinity for its only pole and has no essential 

singularity, is an algebraical polynomial ...... 69 

47. Integral algebraical and integral transcendental functions ... 70 

48. A function, all the singularities of which are accidental, is an algebraical 

meromorphic function .......... 71 



CHAPTER V. 

TRANSCENDENTAL INTEGRAL UNIFORM FUNCTIONS. 

49, 50. Construction of a transcendental integral function with assigned zeros 
a 1? a 2 , a 3 , ..., when an integer s can be found such that 2|a n |~ 8 

is a converging series 74 

51. Weierstrass s construction of a function with any assigned zeros . . 77 
52, 53. The most general form of function with assigned zeros and having 

its single essential singularity at 0=00 . . . . . . 80 

54. Functions with the singly-infinite system of zeros given by ;Z = TO<B, for 

integral values of m 82 

55 57. Weierstrass s o--function with the doubly-infinite system of zeros given 

by z=ma> + m a>, for integral values of TO and of TO . . . . 84 
58. A function cannot exist with a triply-infinite arithmetical system of zeros 88 

59, 60. Class (genre) of a function 89 

61. Laguerre s criterion of the class of a function 91 



CHAPTER VI. 

FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES. 

62. Indefiniteness of value of a function at an essential singularity . . 94 

63. A function is of the form O { =- ) + P (z 6) in the vicinity of an essen- 

\ o/ 

tial singularity at b, a point in the finite part of the plane . . 96 
64, 65. Expression of a function with n essential singularities as a sum of n 

functions, each with only one essential singularity .... 99 
66, 67. Product-expression of a function with n essential singularities and no 

zeros or accidental singularities 101 

68 71. Product-expression of a function with n essential singularities and with 

assigned zeros and assigned accidental singularities ; with a note 

on the region of continuity of such a function . . . .104 



xiv CONTENTS 



CHAPTER VII. 

FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES, 
AND EXPANSION IN SERIES OF FUNCTIONS. 

1>AGE 

72. Mittag-Leffler s theorem on functions with unlimited essential singu 

larities, distributed over the whole plane 112 

73. Construction of subsidiary functions, to be terms of an infinite sum . 113 
74_76. Weierstrass s proof of Mittag-Leffler s theorem, with the generalisation 

of the form of the theorem 114 

77, 78. Mittag-Leffler s theorem on functions with unlimited essential singu 
larities, distributed over a finite circle 117 

79. Expression of a given function in Mittag-Leffler s form .... 123 

80. General remarks on infinite series, whether of powers or of functions . 126 

81. A series of powers, in a region of continuity, represents one and only 

one function ; it cannot be continued beyond a natural limit . . 128 

82. Also a series of functions : but its region of continuity may consist of 

distinct parts 129 

83. A series of functions does not necessarily possess a derivative at points 

on the boundary of any one of the distinct portions of its region 

of continuity ........ 133 

84. A series of functions may represent different functions in distinct parts 

of its region of continuity ; Tannery s series 136 

85. Construction of a function which represents different assigned functions 

in distinct assigned parts of the plane . . . . . .138 

86. Functions with a line of essential singularity 139 

87. Functions with an area of essential singularity or lacunary spaces . 141 

88. Arrangement of singularities of functions into classes and species . . 146 



CHAPTER VIII. 

MULTIFORM FUNCTIONS. 

89. Branch-points and branches of functions 149 

90. Branches obtained by continuation: path of variation of independent 

variable between two points can be deformed without affecting a 
branch of a function if it be not made to cross a branch-point . 150 

91. If the path be deformed across a branch-point which affects the branch, 

then the branch is changed I 55 

92. The interchange of branches for circuits- round a branch-point is cyclical 156 

93. Analytical form of a function near a branch-point 157 

94. Branch-points of a function defined by an algebraical equation in their 

relation to the branches : definition of algebraic function . . 161 
95, 96. Infinities of an algebraic function 163 



CONTENTS xv 

PAGE 



97. Determination of the branch-points of an algebraic function, and of the 

cyclical systems of the branches of the function ... 168 

98. Special case, when the branch-points are simple : their number . . 174 

99. A function, with n branches and a limited number of branch-points and 

singularities, is a root of an algebraical equation of degree n. . 175 



CHAPTER IX. 

PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS 
IN GENERAL. 

100. Conditions under which the path of variation of the integral of a 

multiform function can be deformed without changing the value 

of the integral ....... JQ 

101. Integral of a multiform function round a small curve enclosing a 

branch-point ....... 183 

102. Indefinite integrals of uniform functions with accidental singularities ; 

fdz f dz 

j i 2 ..... ..... 184 



103. Hermite s method of obtaining the multiplicity in value of an integral; 

sections in the plane, made to avoid the multiplicity . . .185 

104. Examples of indefinite integrals of multiform functions ; \wdz round 

any loop, the general value of J(l - 2 2 ) ~ * dz, of J{1 - z 2 ) (1 - k^}} ~ * dz, 

and of 5{(z-e l )(z-e 2 )(g-e s )}-*dz ....... 189 

105. Graphical representation of simply-periodic and of doubly-periodic 

functions ....... 198 

106. The ratio of the periods of a uniform doubly-periodic function is not 

real ............. 200 

107, 108. Triply-periodic uniform functions of a single variable do not exist . 202 

109. Construction of a fundamental parallelogram for a uniform doubly- 

periodic function ....... 205 

110. An integral, with more periods than two, can be made to assume any 

value by a modification of the path of integration between the 
limits ........ 208 



CHAPTER X. 

SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS. 

2rrzi 

111. Simply-periodic functions, and the transformation Z=e w . ; . 211 

112. Fourier s series and simply-periodic functions 213 

113, 114. Properties of simply-periodic functions without essential singularities 

in the finite part of the plane 214 

115. Uniform doubly-periodic functions, without essential singularities in 

the finite part of the plane 218 

116. Properties of uniform doubly-periodic functions 219 



CONTENTS 



117. The zeros and the singularities of the derivative of a doubly-periodic 

function of the second order . . 231 

118, 119. Kelations between homoperiodic functions . . ... . 233 



CHAPTER XL 

DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER. 

120 121. Formation of an uneven function with two distinct irreducible in 
finities; its addition-theorem 243 

122, 123. Properties of Weierstrass s o-function . . . . 247 
124. Introduction of f (2) and of Q(z) 250 

125, 126. Periodicity of the function #> (z), with a single irreducible infinity of 

degree two; the differential equation satisfied by the function #> (2) 251 

127. Pseudo-periodicity of f() 255 

128. Construction of a doubly-periodic function in terms of f (z) and its 

derivatives . - . 256 

129. On the relation qw - j/eo = %iri 25>7 

130. Pseudo-periodicity of a (z) 259 

131. Construction of a doubly-periodic function as a product of o-functions ; 

with examples 259 

132. On derivatives of periodic functions with regard to the invariants 

#2 and 3 * lf K 

133 135. Formation of an even function of either class 266 



CHAPTER XII. 

PSEUDO-PERIODIC FUNCTIONS. 

136. Three kinds of pseudo-periodic functions, with the characteristic equa 
tions 273 

137, 138. Hermite s and Mittag-Leffler s expressions for doubly-periodic functions 

of the second kind 275 

139. The zeros and the infinities of a secondary function . . - . . 280 

140, 141. Solution of Lamp s differential equation 281 

142. The zeros and the infinities of a tertiary function .... 286 

143. Product-expression for a tertiary function 287 

144146. Two classes of tertiary functions ; Appell s expressions for a function 

of each class as a sum of elements 288 

147. Expansion in trigonometrical series 293 

148. Examples of other classes of pseudo-periodic functions . . . 295 



CONTENTS Xvii 

CHAPTER XIII. 

FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM. 
PAGE 

149. Definition of an algebraical addition-theorem 297 

150. A function defined by an algebraical equation, the coefficients of 

which are algebraical functions, or simply-periodic functions, or 
doubly-periodic functions, has an algebraical addition-theorem . 297 

151 154. A function possessing an algebraical addition-theorem is either 
algebraical, simply-periodic or doubly-periodic, having in each 
instance only a finite number of values for an argument . . 300 

155, 156. A function with an algebraical addition-theorem can be defined by a 
differential equation of the first order, into which the independent 
variable does not explicitly enter 309 

CHAPTER XIV. 

CONNECTIVITY OF SURFACES. 

157159. Definitions of connection, simple connection, multiple connection, cross 
cut, loop-cut . . . . .-., 312 

160. Relations between cross-cuts and connectivity 315 

161. Relations between loop-cuts and connectivity 320 

162. Effect of a slit .321 

163, 164. Relations between boundaries and connectivity 322 

165. Lhuilier s theorem on the division of a connected surface into 

curvilinear polygons 325 

166. Definitions of circuit, reducible, irreducible, simple, multiple, compound, 

reconcileable 327 

167, 168. Properties of a complete system of irreducible simple circuits on a 

surface, in its relation to the connectivity 328 

169. Deformation of surfaces 332 

170. Conditions of equivalence for representation of the variable . . 333 



CHAPTER XV. 

RIEMANN S SURFACES. 

171. Character and general description of a Riemann s surface . - .. . 336 

172. Riemann s surface associated with an algebraical equation . . .338 

173. Sheets of the surface are connected along lines, called branch-lines . 338 

174. Properties of branch-lines 340 

175, 176. Formation of system of branch-lines for a surface ; with examples . 341 

177. Spherical form of Riemann s surface . . . 34(5 



XV111 CONTENTS 

PAGE 

178. The connectivity of a Eiemann s surface 347 

179. Irreducible circuits : examples of resolution of Riemann s surfaces 

into surfaces that are simply connected 350 

180, 181. General resolution of a Riemann s surface 353 

182. A Riemann s %-sheeted surface when all the branch-points are simple 355 

183, 184. On loops, and their deformation 356 

185. Simple cycles of Clebsch and Gordan 359 

186 189. Canonical form of Riemann s surface when all the branch -points are 

simple, deduced from theorems of Luroth and Clebsch. . . 361 

190. Deformation of the surface 365 

191. Remark on uniform algebraical transformations 367 



CHAPTER XVI. 

ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS. 

192. Two subjects of investigation 368 

193, 194. Determination of the most general uniform function of position on a 

Riemann s surface .......... 369 

195. Preliminary lemmas in integration on a Riemann s surface . . . 372 

196, 197. Moduli of periodicity for cross-cuts in the resolved surface . . . 373 

198. The number of linearly independent moduli of periodicity is equal to 

the number of cross-cuts, which are necessary for the resolution 

of the surface into one that is simply connected .... 378 

199. Periodic functions on a Riemann s surface, with examples . . . 379 

200. Integral of the most general uniform function of position on a 

Riemann s surface . . . ; 387 

201. Integrals, everywhere finite on the surface, connected with the equa 

tion w*=S(z) 388 

202 204. Infinities of integrals on the surface connected with the algebraical 
equation f (w, z) = 0, when the equation is geometrically interpret- 
able as the equation of a (generalised) curve of the nth order . 388 

205, 206. Integrals of the first kind connected with/(w, z) = 0, Dem g functions 
that are everywhere finite : the number of such integrals, linearly 
independent of one another : they are multiform functions . . 394 

207, 208. Integrals of the second kind connected with f (w, z) = 0, being func 
tions that have only algebraical infinities; elementary integral of 
the second kind .......... 396 

209. Integrals of the third kind connected with/(w, z) = 0, being functions 

that have logarithmic infinities 400 

210, 211. An integral of the third kind cannot have less than two logarithmic 

infinities ; elementary integral of the third kind .... 401 



CONTENTS 



CHAPTER XVII. 

SCHWARZ S PROOF OF THE EXISTENCE-THEOREM. 

PAGE 
212, 213. Existence of functions on a Riemann s surface; initial limitation of 

the problem to the real parts u of the functions . ... . . 405 

214. Conditions to which u, the potential function, is subject . . . 407 

215. Methods of proof : summary of Schwarz s investigation . . . 408 
216 220. The potential-function u is uniquely determined for a circle by the gene 
ral conditions and by the assignment of finite boundary values . 410 

221. Also for any plane area, on which the area of a circle can be con- 

formally represented 423 

222. Also for any plane area which can be obtained by a topological com 

bination of areas, having a common part and each conformally 
representable on the area of a circle 425 

223. Also for any area on a Riemann s surface in which a branch-point 

occurs ; and for any simply connected surface .... 428 
224 227. Real functions exist on a Riemann s surface, everywhere finite, and 

having arbitrarily assigned real moduli of periodicity . . . 430 

228. And the number of the linearly independent real functions thus ob 

tained is 2p ........... 434 

229. Real functions exist with assigned infinities on the surface and 

assigned real moduli of periodicity. Classes of functions of the 
complex variable proved to exist on the Riemann s surface . . 435 



CHAPTER XVIII. 

APPLICATIONS OF THE EXISTENCE-THEOREM. 

230. Three special kinds of functions on a Riemann s surface . . . 437 
231 233. Relations between moduli of functions of the first kind and those of 

functions of the second kind 439 

234. The number of linearly independent functions of the first kind on a 

Riemann s surface of connectivity 2/; + l is p 

235. Normal functions of the first kind ; properties of their moduli . 

236. Normal elementary functions of the second kind : their moduli . 
237, 238. Normal elementary functions of the third kind : their moduli : inter 
change of arguments and parametric points 449 

239. The inversion-problem for functions of the first kind .... 453 

240. Algebraical functions on a Riemann s surface without infinities at the 

branch-points but only at isolated ordinary points on the surface : 
Riemann-Roch s theorem : the smallest number of singularities 
that such functions may possess . . . . . . .457 

241. A class of algebraic functions infinite only at branch-points . . 460 

242. Fundamental equation associated with an assigned Riemann s surface 462 



XX CONTENTS 

PAGE 

243. Appell s factorial functions on a Riemann s surface : their multipliers 

at the cross-cuts 464 

244, 245. Expression for a factorial function with assigned zeros and assigned 
infinities; relations between zeros and infinities of a factorial 
function . . ... . . . . 466 

246. Functions defined by differential equations of the form / ( w, -y- ) = 470 

\ **&) 

247 249. Conditions that the function should be a uniform function of z. . 471 
250, 251. Classes of uniform functions that can be so defined, with criteria of 

discrimination . . , 476 

(dw\ s 
~T~ ) =/ ( w ) .... 482 
az j 



CHAPTER XIX. 

CONFORMAL REPRESENTATION : INTRODUCTORY. 

253. A relation between complex variables is the most general relation that 

secures conformal similarity between two surfaces .... 491 

254. One of the surfaces for conformal representation may, without loss of 

generality, be taken to be a plane 495 

255, 256. Application to surfaces of revolution ; in particular, to a sphere, so 

as to obtain maps .......... 496 

257. Some examples of conformal representation of plane areas, in par 

ticular, of areas that can be conformally represented on the area 

of a circle 501 

258. Linear homographic transformations (or substitutions) of the form 

w = ,: their fundamental properties 512 

cz + d 

259. Parabolic, elliptic, hyperbolic and loxodromic substitutions . . . 517 

260. An elliptic substitution is either periodic or infinitesimal : substitu 

tions of the other classes are neither periodic nor infinitesimal . 521 

261. A linear substitution can be regarded geometrically as the result of 

an even number of successive inversions of a point with regard 

to circles . ......... 523 



CHAPTER XX. 

CONFORMAL REPRESENTATION : GENERAL THEORY. 

262. Riemann s theorem on the conformal representation of a given area 

upon the area of a circle with unique correspondence . . . 525 

263, 264. Proof of Riemann s theorem : how far the functional equation is 

algebraically determinate 526 

265, 266. The method of Beltrami and Cayley for the construction of the 

functional equation for an analytical curve 530 



CONTENTS XXI 

PAGE 

267, 268. Conformal representation of a convex rectilinear polygon upon the 

half-plane of the variable 537 

269. The triangle, and the quadrilateral, conformally represented . . 543 

270. A convex curve, as a limiting case of a polygon .... 548 
271, 272. Conformal representation of a convex figure, bounded by circular arcs : 

the functional relation is connected with a linear differential 
equation of the second order ........ 549 

273. Conformal representation of a crescent ....... 554 

274 276. Conformal representation of a triangle, bounded by circular arcs . 555 
277 279. Relation between the triangle, bounded by circles, and the stereographic 

projection of regular solids inscribed in a sphere .... 563 

280. On families of plane algebraical curves, determined as potential-curves 
by a single parameter u + vi : the forms of functional relation 

), which give rise to such curves .... 575 



CHAPTER XXI. 

GROUPS OF LINEAR SUBSTITUTIONS. 

281. The algebra of group-symbols 582 

282. Groups, which are considered, are discontinuous and have a finite 

number of fundamental substitutions 584 

283, 284. Anharmonic group : group for the modular-functions, and division of 

the plane of the variable to represent the group .... 586 
285, 286. Fuchsian groups : division of plane into convex curvilinear polygons : 

polygon of reference 591 

287. Cycles of angular points in a curvilinear polygon .... 595 

288, 289. Character of the division of the plane : example .... 599 

290. Fuchsian groups which conserve a fundamental circle . . . 602 

291. Essential singularities of a group, and of the automorphic functions 

determined by the group ........ 605 

292, 293. Families of groups : and their class 606 

294. Kleinian groups : the generalised equations connecting two points in 

space "... . . .609 

295. Division of plane and division of space, in connection with Kleinian 

groups 613 

296. Example of improperly discontinuous group 615 



CHAPTER XXII. 

AUTOMORPHIC FUNCTIONS. 

297. Definition of automorphic functions . . . . . - . .619 

298. Examples of functions, automorphic for finite discrete groups of sub 

stitutions 620 

299. Cayley s analytical relation between stereographic projections of posi 

tions of a point on a rotated sphere 620 



XX11 CONTENTS 

PAGE 

300. Polyhedral groups ; in particular, the dihedral group, and the tetra- 

hedral group 623 

301, 302. The tetrahedral functions, and the dihedral functions . . . 628 

303. Special illustrations of infinite discrete groups, from the elliptic 

modular-functions 633 

304. Division of the plane, and properties of the fundamental polygon of 

reference, for any infinite discrete group that conserves a funda 
mental circle ........... 637 

305, 306. Construction of Thetafuchsian functions, pseudo-automorphic for an 

infinite group of substitutions 641 

307. Relations between the number of irreducible zeros and the number 

of irreducible poles of a pseudo-automorphic function, constructed 

with a rational algebraical meromorphic function as element . 645 

308. Construction of automorphic functions ....... 650 

309. The number of irreducible points, for which an automorphic function 

acquires an assigned value, is independent of the value . . 651 

310. Algebraical relations between functions, automorphic for a group : 

application of Riemanu s theory of functions .... 653 

311. Connection between automorphic functions and linear differential 

equations ; with illustrations from elliptic modular-functions . 654 



GLOSSARY OF TECHNICAL TERMS . . . . ; . . , . 659 
INDEX OF AUTHORS QUOTED . . . . - . . . . . 662 
GENERAL INDEX 664 



CHAPTER I. 

GENERAL INTRODUCTION. 

1. ALGEBRAICAL operations are either direct or inverse. Without 
entering into a general discussion of the nature of irrational and of imaginary 
quantities, it will be sufficient to point out that direct algebraical operations 
on numbers that are positive and integral lead to numbers of the same 
character; and that inverse algebraical operations on numbers that are 
positive and integral lead to numbers, which may be negative or fractional 
or irrational, or to numbers which may not even fall within the class of real 
quantities. The simplest case of occurrence of a quantity, which is not 
real, is that which arises when the square root of a negative quantity is 
required. 

Combinations of the various kinds of quantities that may occur are of 
the form x + iy, where x and y are real and i, the non-real element of the 
quantity, denotes the square root of - 1. It is found that, when quantities 
of this character are subjected to algebraical operations, they always lead to 
quantities of the same formal character; and it is therefore inferred that 
the most general form of algebraical quantity is x + iy. 

Such a quantity ic + iy, for brevity denoted by z, is usually called a 
complex variable*; it therefore appears that the complex variable is the 
most general form of algebraical quantity which obeys the fundamental laws 
of ordinary algebra. 

2. The most general complex variable is that, in which the constituents 
x and y are independent of one another and (being real quantities) are 
separately capable of assuming all values from - oo to + oo ; thus a doubly- 
infinite variation is possible for the variable. In the case of a real variable, 
it is convenient to use the customary geometrical representation by measure 
ment of distance along a straight line; so also in the case of a complex 



* The conjugate complex, viz. x - iy, is frequently denoted by z a . 
F. 






2 GEOMETRICAL REPRESENTATION OF [2. 

variable, it is convenient to associate a geometrical representation with 
the algebraical expression ; and this is the well-known representation of 
the variable ac + iy by means of a point with coordinates x and y referred 
to rectangular axes*. The complete variation of the complex variable z 
is represented by the aggregate of all possible positions of the associated 
point, which is often called the point z ; the special case of real variables 
being evidently included in it because, when y = 0, the aggregate of 
possible points is the line which is the range of geometrical variation of 
the real "variable. 

The variation of z is said to be continuous when the variations of x and y 
are contiguous. Continuous variation of z between two given values will 
thus be represented by continuous variation in the position of the point z, 
that is, by a continuous curve (not necessarily of continuous curvature) 
between the points corresponding to the two values. But since an infinite 
number of curves can be drawn between two points in a plane, continuity of 
line is not sufficient to specify the variation of the complex variable ; and, in 
order to indicate any special mode of variation, it is necessary to assign, 
either explicitly or implicitly, some determinate law connecting the variations 
of x and y or, what is the same thing, some determinate law connecting x 
and y. The analytical expression of this law is the equation of the curve 
which represents the aggregate of values assumed by the variable between 
the two given values. 

In such a case the variable is often said to describe the part of the curve 
between the two points. In particular, if the variable resume its initial 
value, the representative point must return to its initial position ; and then 
the variable is said to describe the whole curve -f-. 

When a given closed curve is continuously described by the variable, 
there are two directions in which the description can take place. From 
the analogy of the description of a straight line by a point representing a 
real variable, one of these directions is considered as positive and the other 

* This method of geometrical representation of imaginary quantities, ordinarily assigned to 
Gauss, was originally developed by Argand who, in 1806, published his " Essai sur une maniere 
de representer les quantites imaginaires dans les constructions geometriques." This tract was 
republished in 1874 as a second edition (Gauthier-Villars) ; an interesting preface is added 
to it by Hoiiel, who gives an account of the earlier history of the publications associated with 
the theory. 

Other references to the historical development are given in Chrystal s Text-book of Algebra, 
vol. i, pp. 248, 249; in Holzmiiller s Einfilhrung in die Theorie der isogonalen Venvandschaften 
und dcr conformen Abbildungen, verbunden mit Anwendungen auf mathematische Physik, pp. 1 10, 
21 23 ; in Schlomilch s Compendium der hoheren Analysis, vol. ii, p. 38 (note) ; and in Casorati, 
Teorica delle funzioni di variabili complesse, only one volume of which was published. In this 
connection, an article by Cayley (Quart. Journ. of Math,, vol. xxii, pp. 270 308) may be 
consulted with advantage. 

t In these elementary explanations, it is unnecessary to enter into any discussion of 
the effects caused by the occurrence of singularities in the curve. 



2-] 



THE COMPLEX VARIABLE 




Fig. 1. 



as negative. The usual convention under which one of the directions is 
selected as the positive direction depends upon the conception that the curve 
is the boundary, partial or complete, of some area ; under it, that direction is 
taken to be positive which is such that the bounded area lies to the left of 
the direction of description. It is easy to see that the same direction is taken 
to be positive under an equivalent convention 
which makes it related to the normal drawn 
outwards from the bounded area in the same 
way as the positive direction of the axis of y 
is to the positive direction of the axis of x 
in plane coordinate geometry. 

Thus in the figure (fig. 1), the positive 
direction of description of the outer curve 
for the area included by it is DEF; the 
positive direction of description of the inner 
curve for the area without it (say, the area 
excluded by it) is AGB ; and for the area 
between the curves the positive direction of description of the boundary, 
which consists of two parts, is DEF, ACB. 

3. Since the position of a point in a plane can be determined by means 
of polar coordinates, it is convenient in the discussion of complex variables 
to introduce two quantities corresponding to polar coordinates. 

In the case of the variable z, one of these quantities is (# 2 + y n -)l, the 
positive sign being always associated with it ; it is called the modulus* of 
the variable and it is denoted, sometimes by mod. z, sometimes by \z . 

The other is 0, the angular coordinate of the point z ; it is called the 
argument (and, less frequently, the amplitude) of the variable. It is 
measured in the trigonometrically positive sense, and is determined by 
the equations 

<K=\Z\ cos 6, y= z\ sin#, 

so that z= z\e ei . The actual value depends upon the way in which the 
variable has acquired its value ; when variation 
of the argument is considered, its initial value 
is usually taken to lie between and 2?r or, less 
frequently, between -TT and +TT. 

As z varies in position, the values of \z\ 
and 6 vary. When z has completed a positive 
description of a closed curve, the modulus of z 
returns to the initial value whether the origin Fi g . 2 . 




Der absolute Metro,;) is often used by German writers. 



12 



GREAT VALUES OF 



[3. 



be without, within or on the curve. The argument of z resumes its initial 
value, if the origin (fig. 2) be without the curve ; but, if the origin be 
within the curve, the value of the argument is increased by 2-rr when z 
returns to its initial position. 

If the origin be on the curve, the argument of z undergoes an abrupt 
change by TT as z passes through the origin ; and the change is an increase 
or a decrease according as the variable approaches its limiting position on the 
curve from without or from within. No choice need be made between these 
alternatives; for care is always exercised to choose curves which do not 
introduce this element of doubt. 

4. Representation on a plane is obviously more effective for points at a 
finite distance from the origin than for points at a very great distance. 

One method of meeting the difficulty of representing great values is to 
introduce a new variable z 1 given by z z=\\ the part of the new plane for 
z which lies quite near the origin corresponds to the part of the old plane 
for z which is very distant. The two planes combined give a complete 
representation of variation of the complex variable. 

Another method, in many ways more advantageous, is as follows. Draw 
a sphere of unit diameter, touching the 2-plane at the origin (fig. 3) on 
the under side: join a point z in the plane to , the other extremity of 
the diameter through 0, by a straight line cutting the sphere in Z. 
Then Z is a unique representative of z, that is, a single point on the 
sphere corresponds to a single point on the plane : and therefore the variable 
can be represented on the surface of the sphere. With this mode of 




Fig. 3. 



representation, evidently corresponds to an infinite value of z : and points 
at a very great distance in the 2-plane are represented by points in the 
immediate vicinity of on the sphere. The sphei-e thus has the advantage 
of putting in evidence a part of the surface jn which the variations of 



4.] THE COMPLEX VARIABLE 5 

great values of z can be traced*, and of exhibiting the uniqueness of 
z oo as a value of the variable, a fact that is obscured in the represen 
tation on a plane. 

The former method of representation can be deduced by means of the 
sphere. At draw a plane touching the sphere : and let the straight line 
OZ cut this plane in z . Then z is a point uniquely determined by Z 
and therefore uniquely determined by z. In this new /-plane take axes 
parallel to the axes in the 2-plane. 

The points z and / move in the same direction in space round 00 
as an axis. If we make the upper side of the 2-plane correspond to the 
lower side of the /-plane, and take the usual positive directions in the 
planes, being the positive trigonometrical directions for a spectator looking 
at the surface of the plane in which the description takes place, we have 
these directions indicated by the arrows at and at respectively, so 
that the senses of positive rotations in the two planes are opposite in 
space. Now it is evident from the geometry that Oz and O z are 
parallel ; hence, if be the argument of the point z and & that of the 
point z so that 6 is the angle from Ox to Oz and 6 the angle from O x 
to O z, we have 

6 + ff = ZTT. 

Oz 00 
Further, by similar triangles, -^- t = ^-f , 

that is, Oz . O z = OO 2 = 1. 

Now, if z and z be the variables, we have 

z=0z.e ei , z =0 z .e ffi , 
so that zz =0z.0 z .e^ s ^ 

= 1, 
which is the former relation. 

The /-plane can therefore be taken as the lower side of a plane touching 
the sphere at when the 2-plane is the upper side of a plane touching 
it at 0. The part of the 2-plane at a very great distance is represented on 
the sphere by the part in the immediate vicinity of : and this part of 
the sphere is represented on the /-plane by its portion in the immediate 
vicinity of , which therefore is a space wherein the variations of infinitely 
great values of z can be traced. 

But it need hardly be pointed out that any special method of represent 
ation of the variable is not essential to the development of the theory of 
functions ; and, in particular, the foregoing representation of the variable, 
when it has very great values, merely provides a convenient method of 
dealing with quantities that tend to become infinite in magnitude. 

* This sphere is sometimes called Neumann s sphere; it is used by him for the representation 
of the complex variable throughout his treatise Vorlesungen uber Riemann a Theorie der AlcVschen 
Integrate (Leipzig, Teubner, 2nd edition, 1884). 



6 CONDITIONS OF [5. 

5. The simplest propositions relating to complex variables will be 
assumed known. Among these are, the geometrical interpretation of opera 
tions such as addition, multiplication, root-extraction ; some of the relations 
of complex variables occurring as roots of algebraical equations with real 
coefficients; the elementary properties of functions of complex variables 
which are algebraical and integral, or exponential, or circular functions; 
and simple tests of convergence of infinite series and of infinite products*. 

6. All ordinary operations effected on a complex variable lead, as already 
remarked, to other complex variables; and any definite quantity, thus 
obtained by operations on z, is necessarily a function of z. 

But if a complex variable w be given as a complex function of x 
and y without any indication of its source, the question as to whether 
w is or is not a function of z requires a consideration of the general idea 
of functionality. 

It is convenient to postulate u + iv as a form of the complex variable w, 
where u and v are real. Since w is initially unrestricted in variation, we 
may so far regard the quantities u and v as independent and therefore as 
any functions of x and y, the elements involved in z. But more explicit 
expressions for these functions are neither assigned nor supposed. 

The earliest occurrence of the idea of functionality is in connection with 
functions of real variables ; and then it is coextensive with the idea of 
dependence. Thus, if the value of X depends on that of x and on no other 
variable magnitude, it is customary to regard X as a function of x\ and 
there is usually an implication that X is derived from x by some series of 
operations^. 

A detailed knowledge of z determines x and y uniquely ; hence the values 
of u and v may be considered as known and therefore also w. Thus the 
value of w is dependent on that of z, and is independent of the values 
of variables unconnected with z; therefore, with the foregoing view of 
functionality, w is a function of z. 

It is, however, equally consistent with that view to regard w as a complex 
function of the two independent elements from which z is constituted ; and 
we are then led merely to the consideration of functions of two real 
independent variables with (possibly) imaginary coefficients. 

* These and other introductory parts of the subject are discussed in Chrystal s Text-book of 
Algebra and in Hobson s Treatise on Plane Trigonometry. 

They are also discussed at some length in the recently published translation, by G. L. 
Cathcart, of Harnack s Elements of the differential and integral calculus (Williams and Norgate, 
1891), the second and the fourth books of which contain developments that should be consulted 
in special relation with the first few chapters of the present treatise. 

These books, together with Neumann s treatise.cited in the note on p. 5, will hereafter be cited 
by the names of their respective authors. 

t It is not important for the present purpose to keep in view such mathematical expressions 
as have intelligible meanings only when the independent variable is confined within limits. 



6.] FUNCTIONAL DEPENDENCE 7 

Both of these aspects of the dependence of w on z require that z be 
regarded as a composite quantity involving two independent elements which 
can be considered separately. Our purpose, however, is to regard z as the 
most general form of algebraical variable and therefore as an irresoluble 
entity ; so that, as this preliminary requirement in regard to z is unsatisfied, 
neither of the aspects can be adopted. 

7. Suppose that w is regarded as a function of z in the sense that it can 
be constructed by definite operations on z regarded as an irresoluble 
magnitude, the quantities u and v arising subsequently to these operations 
by the separation of the real and the imaginary parts when z is replaced by 
x + iy. It is thereby assumed that one series of operations is sufficient for 
the simultaneous construction of u and v, instead of one series for u and 
another series for v as in the general case of a complex function in 6. 
If this assumption be justified by the same forms resulting from the two 
different methods of construction, it follows that the two series of opera 
tions, which lead in the general case to u and to v, must be equivalent to 
the single series and must therefore be connected by conditions ; that is, u 
and v as functions of a; and y must have their functional forms related. 

We thus take 

u + iv w = f(z) = f(x + iy) 

without any specification of the form of f. When this postulated equation 

is valid, we have 

dw dw dz ,. , . dw 

_ . _ _ - I { 2/9 TTT _ 

dx dz dx J ^ dz 
dw _ dw "dz _ .,,. . . dw 
frj = ~fad~y~V (Z) l fa 

dw 1 dw dw 

and therefore = - = - ........................... (1) 

dx i dy dz 

equations from which the functional form has disappeared. Inserting the 
value of w, we have 



whence, after equating real and imaginary parts, 

dv _du du _ dv 
dx dy dx dy" 
These are necessary relations between the functional forms of u and v. 

These relations are easily seen to be sufficient to ensure the required 
functionality. For, on taking w = ii + iv, the equations (2) at once lead to 

dw _ 1 dw 
dx i dy 

,, , . dw .dw 

that is, to -- \- 1 - = 0, 

ox dy 



8 RIEMANN S [7. 

a linear partial differential equation of the first order. To obtain the most 
general solution, we form a subsidiary system 

dx _ dy _ dw 
T == T == ~0~* 

It possesses the integrals w, x + iy; and then from the known theory of 
such equations we infer that every quantity w satisfying the equation can be 
expressed as a function of x + iy, i.e., of z. The conditions (2) are thus 
proved to be sufficient, as well as necessary. 

8. The preceding determination of the necessary and sufficient conditions 
of functional dependence is based upon the existence of a functional form ; 
and yet that form is not essential, for, as already remarked, it disappears from 
the equations of condition. Now the postulation of such a form is equivalent 
to an assumption that the function can be numerically calculated for each 
particular value of the independent variable, though the immediate expres 
sion of the assumption has disappeared in the present case. Experience of 
functions of real variables shews that it is often more convenient to use 
their properties than to possess their numerical values. This experience is 
confirmed by what has preceded. The essential conditions of functional 
dependence are the equations (1), and they express a property of the function 

w, viz., that the value of the ratio -r is the same as that of ~- , or, in other 

words, it is independent of the manner in which dz ultimately vanishes by 
the approach of the point z + dz to coincidence with the point z. We are 
thus led to an entirely different definition of functionality, viz. : 

A complex quantity w is a function of another complex quantity z, when 
they change together in such a manner that the value of -, is independent of 
the value of the differential element dz. 

This is Riemann s definition* ; we proceed to consider its significance. 

We have 

dw du + idv 
dz dx + idy 

/du .dv\ dx /du .dv\ du 

__ I __ I n _ I _ _____ I I __ L ^ __ I _ Y. _ 

~~ \dx dxj dx + idy \dy dy/ dx + idy 
Let </> be the argument of dz ; then 




_ 

cos < + 1 sin </> 



* Ges. Werke, p. 5; a modified definition is adopted by him, ib., p. 81. 



8.] DEFINITION OF A FUNCTION 

and therefore 

dw . (du .dv .du dv) ,. {du .dv .du dv 

I I I i n I I I 1 a^4> I J I t In 

7 i ii T" ^ 5 57 f a** i<5 T fc ^~ ~r * 

[da; dx dy dy} (dx dx dy dy 

Since -j is to be independent of the value of the differential element dz, 
dz 

it must be independent of <f> the argument of dz ; hence the coefficient 
of e -2* i n the preceding expression must vanish, which can happen only if 

du _dv dv _ du 
dx dy dx dy " 

These are necessary conditions; they are evidently also sufficient to make 
^ independent of the value of dz and therefore, by the definition, to secure 
that w is a function of z. 

By means of the conditions (2), we have 

dw _ du .dv _dw 
dz dx dx dx 

dw .du dv 1 dw 
and also - = i - [_=_. 

dz dy dy i dy 

agreeing with the former equations (1) and immediately derivable from the 
present definition by noticing that dx and idy are possible forms of dz. 

It should be remarked that equations (2) are the conditions necessary 
and sufficient to ensure that each of the expressions 

udx vdy and vdx + udy 

is a perfect differential a result of great importance in many investigations 
in the region of mathematical physics. 

When the conditions (2) are expressed, as is sometimes convenient, in 
terms of derivatives with regard to the modulus of z, say r, and the 
argument of z, say 0, they take the new forms 

du_ldv dv _ Idu, 

^ ~ 57j > ^~ ^TT. (^)- 

or r dv or r da 

We have so far assumed that the function has a differential coefficient 
an assumption justified in the case of functions which ordinarily occur. But 
functions do occur which have different values in different regions of the 

.z-plane, and there is then a difficulty in regard to the quantity , W at the 

boundaries of such regions ; and functions do occur which, though themselves 
definite in value in a given region, do not possess a differential coefficient at 
all points in that region. The consideration of such functions is not of 
substantial importance at present : it belongs to another part of our subject. 



10 CONFORMAL [8. 

It must not be inferred that, because -j- is independent of the direction 
in which dz vanishes when w is a function of z, therefore -=- has only one 

value. The number of its values is dependent on the number of values of w : 
no one of its values is dependent on dz. 

A quantity, defined as a function by Riemann on the basis of this 
property, is sometimes* called an analytical function; but it seems pre 
ferable to reserve the term analytical in order that it may be associated 
hereafter ( 34) with an additional quality of the functions. 

9. The geometrical interpretation of complex variability leads to impor 
tant results when applied to two variables w and z which are functionally 
related. 

Let P and p be two points in different planes, or in different parts of 
the same plane, representing w and z respectively; and suppose that P and 
p are at a finite distance from the points (if any) which cause discontinuity 
in the relationship. Let q and r be any two other points, z + dz and z + 8z, 
in the immediate vicinity of p ; and let Q and E be the corresponding 
points, w + dw and w + &w, in the immediate vicinity of P. Then 

dw j ^ dw ? 
dw = ^r- dz. bw = -r of, 
dz dz 

the value of ~ being the same for both equations, because, as w is a function 
dz 

of z, that quantity is independent of the differential element of z. Hence 

8w _ Bz 
dw dz 

on the ground that , is neither zero nor infinite at z, which is assumed not 

CL2 

to be a point of discontinuity in the relationship. Expressing all the differ 
ential elements in terms of their moduli and arguments, let 

dz = a-e ei , dw rje^ 1 , 

Sz = oV *, 8w = i)<$\ 
and let these values be substituted in the foregoing relation ; then 

77 tr 

tj a 

$-$ = &-&. 

Hence the triangles QPR and qpr are similar to one another, though 
not necessarily similarly situated. Moreover the directions originally chosen 
for pq and pr are quite arbitrary. Thus it appears that a functional relation 

* Harnack, 84. 



1 11 V - (<M\* a. i^ v 

I <a I I "5 / \ "> 

$a?/ \oyj \dy 



9.] REPRESENTATION OF PLANES 11 

between two complex variables establishes the similarity of the corresponding 
infinitesimal elements of those parts of two planes which are in the immediate 
vicinity of the points representing the two variables. 

The magnification of the w-plane relative to the ^-plane at the corre 
sponding points P and p is the ratio of two corresponding infinitesimal 

lengths, say of QP and qp. This is the modulus of -^ ; if it be denoted by 

m, we have 

2 _ dw 2 
dz 

_ du dv du dv 
dx dy dy dx 

Evidently the quantity m, in general, depends on the variables and 
therefore it changes from one point to another ; hence the functional relation 
between w and z does not, in general, establish similarity of finite parts of 
the two planes corresponding to one another through the relation. 

It is easy to prove that w = az + b, where a and b are constants, is the 
only relation which establishes similarity of finite parts ; and that, with this 
relation, a must be a real constant in order that the similar parts may be 
similarly situated. 

If u + iv = w = <}> (z), the curves u = constant and v = constant cut at 
right angles; a special case of the proposition that, if < (x + iy) = u + v^, 
where A, is a real constant and u, v are real, then u= constant and v= constant 
cut at an angle X. 

The process, which establishes the infinitesimal similarity of two planes 
by means of a functional relation between the variables of the planes, may be 
called the conformal representation of one plane on another*. 

The discussion of detailed questions connected with the conformal representation is 
deferred until the later part of the treatise, principally in order to group all such 
investigations together ; but the first of the two chapters, devoted to it, need not be 
deferred so late and an immediate reading of some portion of it will tend to simplify 
many of the explanations relative to functional relations as they occur in the early 
chapters of this treatise. 

10. The analytical conditions of functionality, under either of the 
adopted definitions, are the equations (2). From them it at once follows that 



8^ + ty* = 

* By Gauss (Ges. Werke, t. iv, p. 262) it was styled conforme Abbildung, the name 
universally adopted by German mathematicians. The French title is representation conforme ; 
and, in England, Cayley has used orthomorphosis or ortliomorphic transformation. 



12 CONDITIONS OF FUNCTIONAL DEPENDENCE [10. 

so that neither the real nor the imaginary part of a complex function can be 
arbitrarily assumed. 

If either part be given, the other can be deduced ; for example, let u be 
given ; then we have 



7 j j 

dv = ^-dx + dy 

dx dy 

du , du j 
= -=-dx+~-dy, 
dy ox 

and therefore, except as to an additive constant, the value of v is 

[i 9w 7 du -, \ 

- dx + 5- dy I . 
A dy ax I 

In particular, when u is an integral function, it can be resolved into the 
sum of homogeneous parts 

MI + w 2 + w 3 + . . . ; 

and then, again except as to an additive constant, v can similarly be 
expressed in the form 

V l + V 2 + V 3 + ---- 

It is easy to prove that 

du m du m 

> = y-te-*-ty> 

by means of which the value of v can be obtained. 

The case, when u is homogeneous of zero dimensions, presents no 
difficulty ; for we then have 



v = c-a\ogr, =c-/f 
where a, 6, c are constants. 

Similarly for other special cases; and, in the most general case, only 
a quadrature is necessary. 

The tests of functional dependence of one complex on another are of 
effective importance in the case when the supposed dependent complex 
arises in the form u + iv, where u and v are real; the tests are, of course, 
superfluous when w is explicitly given as a function of z. When w does 
arise in the form u + iv and satisfies the conditions of functionality, perhaps 
the simplest method (other than by inspection) of obtaining the explicit 
expression in terms of z is to substitute z iy for x in u + iv ; the simplified 
result must be a function of z alone. 

11. Conversely, when w is explicitly given as a function of z and it is 
divided into its real and its imaginary parts, these parts individually 
satisfy the foregoing conditions attaching to u and v. Thus logr, where r 
is the distance of a point z from a point a, is the real part of log (z a) 
and therefore satisfies the equation 



11.] 



EXAMPLE OF RIEMANN S DEFINITION 



13 



Again, <f>, the angular coordinate of z relative to the same point a, is 
the real part of i log (z a) and satisfies the same equation : the more 
usual form of < being tan" 1 {(y y )/( %o)}> where a = x + iy . Again, if 
a point z be distant r from a and r from b, then log (r/r \ being the real 
part of log {(z a)l(z b)\, is a solution of the same equation. 

The following example, the result of which will be useful subsequently*, uses the 
property that the value of the derivative is independent of the differential element. 

z-c 



Consider a function 



u + iv = w = log 



where c is the inverse of c with regard to a circle centre the origin and radius R. 
Then 



z-c 

* V 

: r> 

z-c 



and the curves u = constant are circles. Let 

W- 
(fig. 4) Oc = r, xOc = a so that c = re at , c = e al ; 

then if 




Fig. 4. 

the values of X for points in the interior of the circle of radius R vary from zero, when 
circle u = constant is the point c, to unity, when the circle u = constant is the circle of 
radius R. Let the point K ( = 6e al ) be the centre of the circle determined by a value of 
X, and let its radius be p ( = %MN}. Then since 

cM r ,. cN 



we have 



whence 



r+p-d r d + P~ r 

Vp-B Q-p 

r r 



P = 



Now if dn be an element of the normal drawn inwards at z to the circle NzM, we have 
dz = dx+idy= dn . cos ^ - idn . sin ^ 

--*<*, 

where ^ ( = zKx } is the argument of z relative to the centre of the circle. Hence, since 

dw 1 1 



we have 
But 

so that 

and 



, ., , du .dv 

and therefore -=- + i -j- = 
dn dn 



__ _ 

dz z c z-c 1 
du .dv dw 
dn 



.dv dw /I 1 \ ty 

dn dn \z c z c) 



e^ - Re ai ) 



J> _ 1 _ 1 !_ 

I /i! ~\ ^^ 7? * 1 ^ X ff */^ 
\ A7*6 J\G ./t6 i 



* In 217, in connection with the investigations of Schwarz, by whom the result is stated, 
Ges. Werke, t. ii, p. 183. 



14 DEFINITIONS [11. 

Hence, equating the real parts, it follows that 

du (_R 2 -r 2 A 2 ) 2 



dn ~ \R(R*- r 2 ) {E 2 - 2Rr\ cos (^ - Q) + XV 2 } 
the differential element dn being drawn inwards from the circumference of the circle. 

The application of this method is evidently effective when the curves u = constant, 
arising from a functional expression of w in terms of z, are a family of non-intersecting 
algebraical curves. 

12. As the tests which are sufficient and necessary to ensure that a 
complex quantity is a function of z have been given, we shall assume that 
all complex quantities dealt with are functions of the complex variable 
( 6, 7). Their characteristic properties, their classification, and some of 
the simpler applications will be considered in the succeeding chapters. 

Some initial definitions and explanations will now be given. 

(i). It has been assumed that the function considered has a differential 
coefficient, that is, that the rate of variation of the function in any direction 
is independent of that direction by being independent of the mode of change 
of the variable. We have already decided ( 8) not to use the term analytical 
for such a function. It is often called monogenic, when it is necessary to 
assign a specific name ; but for the most part we shall omit the name, the 
property being tacitly assumed*. 

We can at once prove from the definition that, when the derivative 

/ dw\ .-. , if- c <- v dw Idw 

w, = -p- exists, it is itselt a Junction, .bor w-, =-= = - = are equations 
\ dz ) dx i dy 

which, when satisfied, ensure the existence of w^ ; hence 

1 dw-! _ 1 3 (dw\ 
i dy i dy \d% ) 
_ d_ (I dw\ 
dx \i dyj 
_dw 1 
= l)x 

shewing, as in 8, that the derivative ~ is independent of the direction in 

CL2 

which dz vanishes. Hence w l is a function of z. 

Similarly for all the derivatives in succession. 

(ii). Since the functional dependence of a complex is ensured only if the 
value of the derivative of that complex be independent of the manner in 
which the point z + dz approaches to coincidence with z, a question naturally 

* This is in fact done by Biemann, who calls such a dependent complex simply a function. 
Weierstrass, however, has proved ( 85) that the idea of a monogenic function of a complex 
variable and the idea of dependence expressible by arithmetical operations are not coextensive. 
The definition is thus necessary; but the practice indicated in the text will be adopted, as non- 
monogenic functions will be of relatively rare occurrence. 



12.] DEFINITIONS 15 

suggests itself as to the effect on the character of the function that may be 
caused by the manner in which the variable itself has come to the value of z. 
If a function have only one value for each given value of the variable, 
whatever be the manner in which the variable has come to that value, the 
function is called uniform*. Hence two different paths from a point a to a 
point z give at z the same value for any uniform function ; and a closed 
curve, beginning at any point and completely described by the ^-variable, 
will lead to the initial value of w, the corresponding w-curve being closed, if z 
have passed through no point which makes w infinite. 

The simplest class of uniform functions is constituted by algebraical 
rational functions. 

(iii). If a function have more than one value for any given value of the 
variable, or if its value can be changed by modifying the path in which 
the variable reaches that given value, the function is called multiform-] . 
Characteristics of curves, which are graphs of multiform functions corre 
sponding to a 2-curve, will hereafter be discussed. 

One of the simplest classes of multiform functions is constituted by 
algebraical irrational functions. 

(iv). A multiform function has a number of different values for the same 
value of z, and these values vary with z : the aggregate of the variations of 
any one of the values is called a branch of the function. Although the 
function is multiform for unrestricted variation of the variable, it often 
happens that a branch is uniform when the variable is restricted to 
particular regions in the plane. 

(v). A point in the plane, at which two or more branches of a multiform 
function assume the same value, is called a branch-point^ of the function; 
the relations of the branches in the immediate vicinity of a branch-point will 
hereafter be discussed. 

(vi). A function which is monogenic, uniform and continuous over any 
part of the ^-plane is called holomorphic over that part of the plane. When 
a function is called holomorphic without any limitation, the usual implication 
is that the character is preserved over the whole of the plane which is not at 
infinity. 

The simplest example of a holomorphic function is a rational integral 
algebraical polynomial. 

* Also monodromic, or monotropic; with German writers the title is eindeutig, occasionally, 
einandrig. 

t Also polytropic ; with German writers the title is mchrdeittig. 

J Also critical point, which, however, is sometimes used to include all special points of a 
function ; with German writers the title is Verziveigungspunkt, and sometimes Windungspunkt. 
French writers use point de ramification, and Italians punto di giramento and punto di 
diramazione. 

Also synectic. 



16 EXAMPLES ILLUSTRATING [12. 

(vii). A root (or a zero) of a function is a value of the variable for which 
the function vanishes. 

The simplest case of occurrence of roots is in a rational integral alge 
braical function, various theorems relating to which (e.g., the number of 
roots included within a given contour) will be found in treatises on the 
theory of equations. 

(viii). The infinities of a function are the points at which the value of 
the function is infinite. Among them, the simplest are the poles* of the 
function, a pole being an infinity such that in its immediate vicinity the 
reciprocal of the function is holomorphic. 

Infinities other than poles (and also the poles) are called the singular 
points of the function : their classification must be deferred until after the 
discussion of properties of functions. 

(ix). A function which is monogenic, uniform and, except at poles, 
continuous, is called a meromorphic function f. The simplest example is a 
rational algebraical fraction. 

13. The following functions give illustrations of some of the preceding 
definitions. 

(a) In the case of a meromorphic function 

F(z) 
111 * - 

/<*) 

where F and / are rational algebraical functions without a common factor, 
the roots are the roots of F (z) and the poles are the roots of f (z). Moreover, 
according as the degree of F is greater or is less than that of f,z = vo is a 
pole or a zero of w. 

(b) If w be a polynomial of order n, then each simple root of w is a 

branch-point and a zero of w m , where m is a positive integer ; z = oo is 
a pole of w; and z= oo is a pole but not a branch-point or is an infinity 
(though not a pole) and a branch-point of w$ according as n is even or odd. 

(c) In the case of the function 

1 



w- 



sn- 

z 



(the notation being that of Jacobian elliptic functions), the zeros are given by 



z 
for all positive and negative integral values of m and of m . If we take 



- = iK + 2mK + Zm iK -f 

z 

* Also polar discontinuities ; also ( 32) accidental singularities. 

t Sometimes rey-nlar, but this term will be reserved for the description of another property of 
functions. 



13.] THE DEFINITIONS 17 

where may be restricted to values that are not large, then 

w = (- l) m &sn 

so that, in the neighbourhood of a zero, w behaves like a holomorphic 
function. There is evidently a doubly-infinite system of zeros: they are 
distinct from one another except at the origin, where an infinite number 
practically coincide. 

The infinities of w are given by 



for all positive and negative integral values of n and of n . If we take 



- = 2nK + Zn iK + 

2! 

then - = (-l)"sn 

w 

so that, in the immediate vicinity of f=0, - is a holomorphic function. 

Hence f = is a pole of w. There is thus evidently a doubly-infinite system 
of poles ; they are distinct from one another except at the origin, where an 
infinite number practically coincide. But the origin is not a pole; the 
function, in fact, is there not determinate, for it has an infinite number of 
zeros and an infinite number of infinities, and the variations of value are not 
necessarily exhausted. 

For the function j , the origin is a point which will hereafter be called 

sn- 

z 

an essential singularity. 






F. 



CHAPTER II. 

INTEGRATION OF UNIFORM FUNCTIONS. 

14. THE definition of an integral, that is adopted when the variables 
are complex, is the natural generalisation of that definition for real variables 
in which it is regarded as the limit of the sum of an infinite number of 
infinitesimally small terms. It is as follows : 

Let a and z be any two points in the plane ; and let them be connected 
by a curve of specified form, which is to be the path of variation of the 
independent variable. Let f(z) denote any function of 0; if any infinity 
of f(z) lie in the vicinity of the curve, the line of the curve will be chosen 
so as not to pass through that infinity. On the curve, let any number of 
points z^ z 2 ,..., z n in succession be taken between a and z ; then, if the sum 

(z, - a)f (a) + (z, - z,} f (z,) + ... + (z- z n )f(z n } 

have a limit, when n is indefinitely increased so that the infinitely numerous 
points are in indefinitely close succession along the whole of the curve from a 
to z, that limit is called the integral of / (z) between a and z. It is denoted, 
as in the case of real variables, by 



f(z)dz. 

The limit, as the value of the integral, is associated with a particular 
curve : in order that the integral may have a definite value, the curve (called 
the path of integration) must, in the first instance, be specified*. The 
integral of any function whatever may not be assumed to depend in general 
only upon the limits. 

15. Some inferences can be made from the definition. 
(I.) The integral along any path from a to z passing through a point is 
the sum of the integrals from a to and from \ to z along the same path. 

* This specification is tacitly supplied when the variables are real : the variable point moves 
along the axis of x. 



15.] INTEGRATION 19 

Analytically, this is expressed by the equation 

P / (*) dz = I V (*) dz + I V (*) <fc, 

^ a J a J f 

the paths on the right-hand side combining to form the path on the left. 

(II.) When the path is described in the reverse direction, the sign of the 
integral is changed : that is, 



the curve of variation between a and z being the same. 

(III.) The integral of the sum of a finite number of terms is equal to 
the sum of the integrals of the separate terms, the path of integration being 
the same for all. 

(IV.) If a function f (z) be finite and continuous along any finite line 
between two points a and z, the integral \ f(z)dz is finite. 

J a 

Let 7 denote the integral, so that we have I as the limit of 



r=0 

hence |/| = limit of 



Because f(z} is finite and continuous, its modulus is finite and therefore 
must have a superior limit, say M, for points on the line. Thus 



80 that I/I < limit of r+1 

<MS, 

where 8 is the finite length of the path of integration. Hence the modulus 
of the integral is finite ; the integral itself is therefore finite. 

No limitation has been assigned to the path, except finiteness in length ; 
the proposition is still true when the curve is a closed curve of finite length. 

Hermite and Darboux have given an expression for the integral which 
leads to the same result. We have as above 



f(z)\ dz\, 
where 6 is a real positive quantity less than unity. The last integral involves 



22 



20 THEOREMS [15. 

only real variables; hence* for some point lying between a and z, we have 



f 

J a 



so that l/| = fl9f|/(!)|. 

It therefore follows that there is some argument a such that, if X = Be 10 -, 



This form proves the finiteness of the integral ; and the result is the 
generalisation f to complex variables of the theorem just quoted for real 
variables. 

(V.) When a, function is expressed in the form of a series, which converges 
uniformly and unconditionally, the integral of the function along any path of 
finite length is the sum of the integrals of the terms of the series along the 
same path, provided that path lies within the circle of convergence of the series : 
a result, which is an extension of (III.) above. 

Let M + MI + u. 2 + . . . be the converging series ; take 

/ (z) = U + M! + . . . + U n + R, 

where \R\ can be made infinitesimally small with indefinite increase of n, 
because the series converges uniformly and unconditionally. Then by (III.), 
or immediately from the definition of the integral, we have 

rz rs rz rz re 

f(z)dz= I u dz + ^dz + . .. + I u n dz + 1 Rdz, 

J a J a J a J a J a 

the path of integration being the same for all the integrals. Hence, if 

re n re 

(S) = I f (z) dz 2 I u m dz, 

J a m=oJ a 

ft 

we have = I Rdz. 



ft 

= I 

J a 



Let R be the greatest value of \R\ for points in the path of integration 
from a to z, and let 8 be the length of this path, so that 8 is finite ; 

then, by (IV.), 

\\<SR. 

Now 8 is finite ; and, as n is increased indefinitely, the quantity R tends 
towards zero as a limit for all points within the circle of convergence and 
therefore for all points on the path of integration provided that the path lie 
within the circle of convergence. When this proviso is satisfied, |@| becomes 
infinitesimally small and therefore also becomes infinitesimally small with 

* Todhunter s Integral Calculus (4th ed.), 40; Williamson s Integral Calculus, (Gth ed.), 96. 
t Hermite, Cours d la faculte dcs sciences de Paris (4 6mc ed., 1891), p. 59, where the reference 
to Darboux is given. 



15.] 



ON INTEGRATION 



21 



indefinite increase of n. Hence, under the conditions stated in the enuncia 
tion, we have 

rs oo r% 

f(z)dz- 2 I u m dz = 0, 

J a m^QJ a 

which proves the proposition. 

16. The following lemma* is of fundamental importance. 

Let any region of the plane, on which the ^-variable is represented, be 
bounded by one or more simple^ curves which do not meet one another: 
each curve that lies entirely in the finite part of the plane will be considered 
to be a closed curve. 

If p and q be any two functions of cc and y, which, for all points within the 
region or along its boundary, are uniform, finite and continuous, then the 
integral 

fffdq dp\j , 
1 1 a - a dxdy, 

JJ \dx dyj 

extended over the whole area of the region, is equal to the integral 

f(pdx + qdy), 
taken in a positive direction round the whole boundary of the region. 

(As the proof of the proposition does not depend on any special form of 
region, we shall take the area to be (fig. 5) that which is included by the 
curve QiPiQs Pa and excluded by P^Qz PsQs and excluded by P/P 2 . The 
positive directions of description of the curves are indicated by the arrows ; 
and for integration in the area the positive directions are those of increas 
ing a; and increasing y.) 




AB 



Fig. 5. 

* It is proved by Eiemann, Ges. Werke, p. 12, and is made by him (as also by Cauchy) the 
basis of certain theorems relating to functions of complex variables. 

t A curve is called simple, if it have no multiple points. The aim, in constituting the boundary 
from such curves is to prevent the superfluous complexity that arises from duplication of area on 
the plane. If, in any particular case, multiple points existed, the method of meeting the difficulty 
would be to take each simple loop as a boundary. 






22 FUNDAMENTAL THEOREM [16. 

First, suppose that both p and q are real. Then, integrating with regard 
to x, we have * 



where the brackets imply that the limits are to be introduced. When the 
limits are introduced along a parallel GQ^... to the axis of x, then, since 
CQiQi . gives the direction of integration, we have 

[qdy] = - qjdyj. + qi dt/i - q. 2 dy 2 + q- 2 dy 2 - q 3 dy 3 + qdy 9 , 

where the various differential elements are the projections on the axis of y 
of the various elements of the boundary at points along GQiQJ.... 

Now when integration is taken in the positive direction round the whole 
boundary, the part of / qdy arising from the elements of the boundary at the 
points on CQjQ/... is the foregoing sum. For at Q 3 it is q a dy 3 because the 
positive element dy 9 , which is equal to CD, is in the positive direction of 
boundary integration; at Q 3 it is q 3 dy s because the positive element dy 3 , 
also equal to CD. is in the negative direction of boundary integration ; 
at Qz it is q 2 dy 2 , for similar reasons ; at Q. 2 it is q 2 dy a , for similar reasons ; 
and so on. Hence 



corresponding to parallels through C and D to the axis of x, is equal to 
the part of fqdy taken along the boundary in the positive direction for all 
the elements of the boundary that lie between those parallels. Then when 
we integrate for all the elements CD by forming f[qdy], an equivalent is 
given by the aggregate of all the parts of fqdy taken in the positive direction 
round the whole boundary ; and therefore 



on the suppositions stated in the enunciation. 
Again, integrating with regard to y, we have 



when the limits are introduced along a parallel RP^P^. . . to the axis of y : 
the various differential elements are the projections on the axis of x of the 
various elements of the boundary at points along SPjP/.... 

It is proved, in the same way as before, that the part of - jpdx arising 
from the positively-described elements of the boundary at the points on 
BP^ ... is the foregoing sum. At P 3 the part of fpdac is - p 3 dx 3 , because 
the positive element dx 3 , which is equal to AB, is in the negative direction 

* It is in this integration, and in the corresponding integration for p, that the properties of 
the function q are assumed : any deviation from uniformity, finiteness or continuity within the 
region of integration would render necessary some equation different from the one given in 
the text. 



16.] IN INTEGRATION 23 

of boundary integration ; at P 3 it is p 3 dx 3 , because the positive element 
dx 3 , also equal to AB, is in the positive direction of boundary integration; 
and so on for the other terms. Hence 

- [pdas], 

corresponding to parallels through A and B to the axis of y, is equal to 
the part of fpdx taken along the boundary in the positive direction for all 
the elements of the boundary that lie between those parallels. Hence 
integrating for all the elements AB, we have as before 

[[dp j j , j 

~ dxdy = I pax, 

JJdy 

and therefore II U ?r ) dxdy=f(pdx + qdy). 

Secondly, suppose that p and q are complex. When they are resolved 
into real and imaginary parts, in the forms p + ip" and q + iq" respectively, 
then the conditions as to uniformity, finiteness and continuity, which apply to 
p and q, apply also to p , q , p", q". Hence 



and ~ - - dxdy = j(p"dx + q"dy), 

and therefore 1 1 [ 2* _ J 9 j dxdy = J(pdx + qdy} 

JJ \ox oy/ 

which proves the proposition. 

No restriction on the properties of the functions p and q at points 
that lie without the region is imposed by the proposition. They may have 
infinities outside, they may cease to be continuous at outside points or they 
may have branch-points outside ; but so long as they are finite and continuous 
everywhere inside, and in passing from one point to another always acquire 
at that other the same value whatever be the path of passage in the region, 
that is, so long as they are uniform in the region, the lemma is valid. 

17. The following theorem due to Cauchy* can now be proved : _ 
If a function f(z) be holomorphic throughout any region of the z-plane, 
then the integral ff(z) dz, taken round the whole boundary of that region, is zero. 
We apply the preceding result by assuming 

p=f(z\ q = ip = if(z); 

owing to the character of f(z), these suppositions are consistent with the 

* For an account of the gradual development of the theory and, in particular, for a 
statement of Cauchy s contributions to the theory (with references), see Casorati, Teorica 
delle funzioni di variabili complcsse, pp. 64-90, 102-106. The general theory of functions, 
as developed by Briot and Bouquet in their treatise Theoric des fonctiom ellipUques, is based 
upon Cauchy s method. 



24 INTEGRATION OF [17. 

conditions under which the lemma is valid. Since p is a function of z, we 
have, at every point of the region, 

dp _ I dp 

das i dy 
and therefore, in the present case, 

dq _ . dp _ dp 

das doc dy 

There is no discontinuity or infinity of p or q within the region ; hence 



the integral being extended over the region. Hence also 

!(pdx + qdy) = 0, A^ ^/ 
when the integral is taken round the whole boundary of the region. But 

pdx + qdy = pdx + ipdy 
pdz 
=f(z)dz, 

and therefore //(X) dz = 0, 

the integral being taken round the whole boundary of the region within 
which f(z) is holomorphic. 

It should be noted that the theorem requires no limitation on the cha 
racter of/(^) for points z that are not included in the region. 

Some important propositions can be derived by means of the theorem, as 
follows. 

18. When a function f (z) is holomorphic over any continuous region 

rz 
of the plane, the integral I f(z)dz is a holomorphic function of 2 provided the 

J a 

points z and a as well as the whole path of integration lie within that region. 

The general definition ( 14) of an integral is associated with a specified 
path of integration. In order to prove that the integral is a holomorphic 
function of z, it will be necessary to prove (i) that the integral acquires the 
same value in whatever way the point z is attained, that is, that the value is 
independent of the path of integration, (ii) that it is finite, (iii) that it 
is continuous, and (iv) that it is monogenic. 

Let two paths ayz and afiz between a and z be drawn (fig. 6) in the 
continuous region of the plane within which f(z) is 
holomorphic. The line ayzfia is a contour over the area 
of which / (z} is holomorphic ; and therefore ff(z) dz 
vanishes when the integral is taken along ayzfta. 
Dividing the integral into two parts and implying by 
Zy, Zp that the point z has been reached by the paths a" 
a<yz, a{3z respectively, we have Fig. 6. 




18.] HOLOMOEPHIC FUNCTIONS 25 



and therefore */ ( z ) dz = - f (z) dz 

J a J Zg 

-?/*/(*)* 

J a 
Thus the value of the integral is independent of the way in which z has 

FZ 

acquired its value ; and therefore I f(z) dz is uniform in the region. Denote 
it by F(z). 

Secondly, f(z) is finite for all points in the region and, after the result 
of 17, we naturally consider only such paths between a and z as are finite in 
length, the distance between a and z being finite; hence ( 15, IV.) the 
integral F (z} is finite for all points z in the region. 

Thirdly, let z (= z 4- 82) be a point infinitesimally near to z ; and consider 
I f(z) dz. By what has just been proved, the path from a to z can be taken 

J d 

aftzz ; therefore 

(*/(*) dz = [/(z) dz + l Z f(z) dz 

J * J a J z 

fz+8z r z rz+Sz 

or f(z}dz- \ f(z)dz=\ f(z)dz, 

J a J a J z 

fz+Sz 

80 that F(z + Sz) - F(z) = f(z} dz. 

J 2 

Now at points in the infinitesimal line from z to z , the value of the 
continuous function f(z) differs only by an infinitesimal quantity from its 
value at z ; hence the right-hand side is 



where e| is an infinitesimal quantity vanishing with ck It therefore follows 
that 



is an infinitesimal quantity with a modulus of the same order of small 
quantities as \Sz\. Hence F (z) is continuous for points z in the region. 
Lastly, we have 



and therefore F(z + Sz)-F(z) 

82 

has a limit when Sz vanishes; and this limit, f(z), is independent of the 
way in which 8z vanishes. Hence F (z) has a differential coefficient ; the 
integral is monogenic for points z in the region. 






26 INTEGRATION OF [18. 

Hence F (z), which is equal to 

* f(z)d* t 



is uniform, finite, continuous and monogenic; it is therefore a holomorphic 
function of z. 

As in 16 for the functions p and q, so here for f(z), no restriction is 
placed on properties of / (z) at points that do not lie within the region; so 
that elsewhere it may have infinities, or discontinuities or branch points. 
The properties, essential to secure the validity of the proposition, are 
(i) that no infinities or discontinuities lie within the region, and (ii) that the 
same value of f(z) is acquired by whatever path in the continuous region 
the variable reaches its position z. 

COROLLARY. No change is caused in the value of the integral of a 
holomorphic function between two points when the path of integration between 
the points is deformed in any manner, provided only that, during the defor 
mation, no part of the path passes outside the boundary of the region within 
which the function is holomorphic. 

This result is of importance, because it permits special forms of the path 
of integration without affecting the value of the integral. 

19. When a function f(z) is holomorphic over a part of the plane 
bounded by two simple curves (one lying within the other), equal values of 
ff(z) dz are obtained by integrating round each of the curves in a direction, 
which relative to the area enclosed by each is positive. 

The ring-formed portion of the plane (fig. 1, p. 3) which lies between 
the two curves being a region over which f(z) is holomorphic, the integral 
ff(z) dz taken in the positive sense round the whole of the boundary of 
the included portion is zero. The integral consists of two parts : first, that 
round the outer boundary the positive sense of which is DEF , and second, 
that round the inner boundary the positive sense of which for the portion of 
area between ABC and DEF is ACE. Denoting the value of ff(z)dz round 
DEF by (DEF), and similarly for the other, we have 

(ACB) + (DEF) = 0. 

The direction of an integral can be reversed if its sign be changed, so that 
(ACB) = - (ABC) ; and therefore 

(ABC) = (DEF). 

But (ABC) is the integral ff(z)dz taken round ABC, that is, round the 
curve in a direction which, relative to the area enclosed by it, is positive. 

The proposition is therefore proved. 

The remarks made in the preceding case as to the freedom from limitations 
on the character of the function outside the portion are valid also in this case. 



19.] HOLOMORPHIC FUNCTIONS 27 

COROLLARY I. When the integral of a function is taken round the whole 
of any simple curve in the plane, no change is caused in its value by continuously 
deforming the curve into any other simple curve provided that the function 
is holomorphic over the part of the plane in which the deformation is effected. 

COROLLARY II. When a function f (z) is holomorphic over a continuous 
portion of a plane bounded by any number of simple non-intersecting curves, 
all but one of which are external to one another and the remaining one of 
which encloses them all, the value of the integral jf(z) dz taken positively round 
the single external curve is equal to the sum of the values taken round each of 
the other curves in a direction which is positive relative to the area enclosed 
by it. 

These corollaries are of importance in finding the value of the integral 
of a meromorphic function round a curve which encloses one or more of the 
poles. The fundamental theorem for such integrals, also due to Cauchy, is 
the following. 

20. Let f(z) denote a function which is holomorphic over any region in 
the z-plane and let a denote any point within that region, which is not a zero 

ff( 2 ); then 

., , 1 f/0) , 

f( a ) = ^ *-*-* az > 

2vnJ z-a 



the integral being taken positively round the whole boundary of the region. 

With a as centre and a very small radius p, describe a circle G, which will 
be assumed to lie wholly within the region; this assumption is justifiable 
because the point a lies within the region. Because f (z) is holomorphic over 
the assigned region, the f unction f(z)l(z a) is holomorphic over the whole of 
the region excluded by the small circle C. Hence, by Corollary II. of 19, we 
have 



z-a 



the notation implying that the integrations are taken round the whole 
boundary B and round the circumference of G respectively. 

For points on the circle C, let z a = pe ei , so that 9 is the variable for 
the circumference and its range is from to 2?r ; then we have 

dz 



z a 



= id6. 



Along the circle f(z)=f (a + pe ei ) ; the quantity p is very small and / is 
finite and continuous over the whole of the region so that f(a + pe ei ) differs 
from /() only by a quantity which vanishes with p. Let this difference 
be e, which is a continuous small quantity; then |ej is a small quantity 
which, for every point on the circumference of C, vanishes with p. Then 



28 INTEGRATION OF [20. 



" edO. 
o 



If E denote the value of the integral on the right-hand side, and 77 the 
greatest value of the modulus of e along the circle, then, as in 15, 

/2ir 

i E < I e d6 



f 



Now let the radius of the circle diminish to zero: then 77 also diminishes 
to zero and therefore E , necessarily positive, becomes less than any finite 
quantity however small, that is, E is itself zero; and thus we have 



z a 
which proves the theorem. 

This result is the simplest case of the integral of a meromorphic 

f(z} 
function. The subject of integration is , a function which is monogenic 

and uniform throughout the region and which, everywhere except at z = a, is 
finite and continuous ; moreover, z = a is a pole, because in the immediate 

Z ~~~ CL 

vicinity of a the reciprocal of the subject of integration, viz. ^-rr > i- B hl- 
morphic. 

The theorem may therefore be expressed as follows : 

If g (z) be a meromorphic function, which in the vicinity of a can be 

f(z} 
expressed in the form J where f(a) is not zero and which at all other 

Z CL 

points in a region enclosing a is holomorphic, then 

- . fg (z) dz = limit of (z a)g (z) when z a, 



the integral being taken round a curve in the region enclosing the point a. 

The pole a of the function g (z) is said to be simple, or of the first order, 
or of multiplicity unity. 

Corollary. The more general case of a meromorphic function with a 
finite number of poles can easily be deduced. Let these be a 1} ..., a n each 
assumed to be simple ; and let 

G (z) = (z- a,) (z - a a ). ..(z - a n ). 



20.] MEROMORPHIC FUNCTIONS 29 

Let f(z) be a holomorphic function within a region of the 2-plane bounded 
by a simple contour enclosing the n points a 1} a,...a n , no one of which is a 
zero off(z). Then since 



f(z) 1 f(z) 

we have j^~( = S ,, . -^-- . 

6r (#) r= i Or (a r ) z a r 

w ^ f u f/(*),j 3 ! f/(*) ,7 

We therefore have "L , dz = 2< >.. , . I dz, 

J &(*) r =iCr (a r )J 2-a r 

each integral being taken round the boundary. But the preceding proposition 
gives 



because f(z) is holomorphic over the whole region included in the contour ; 
and therefore 



the integral on the left-hand side being taken in the positive direction*. 

The result just obtained expresses the integral of the meromorphic 
function round a contour which includes a finite number of its simple poles. 
It can be otherwise obtained by means of Corollary II. of 19, by adopting 
a process similar to that adopted above, viz., by making each of the curves in 
the Corollary quoted small circles round the points Oj,..., a n with ultimately 
vanishing radii. 

21. The preceding theorems have sufficed to evaluate the integral of 
a function with a number of simple poles : we now proceed to obtain 
further theorems, which can be used among other purposes to evaluate 
the integral of a function with poles of order higher than the first. 

We still consider a function f(z) which is holomorphic within a given 
region. Then, if a be a point within the region which is not a zero of f(z), 
we have 



z - a 



the point a being neither on the boundary nor within an infinitesimal 
distance of it. Let a + Sa be any other point within the region ; then 

dz, 



z a 8a 

* We shall for the future assume that, if no direction for a complete integral be specified, the 
positive direction is taken. 



30 

and therefore 



PROPERTIES OF 



[21. 



iff, 



8a 



f(z)dz, 



t J ((* - a) 2 (z - of (z-a -Sa)j 
the integral being in every case taken round the boundary. 

Since f(z) is monogenic, the definition of / (a), the first derivative of 
/(a), gives / (a) as the limit of 

f(a + Ba)-f(a) 
Ba 

when Ba ultimately vanishes ; hence we may take 



where a is a quantity which vanishes with Ba and is therefore such that \ a \ 
also vanishes with Ba. Hence 



dividing out by Sa and transposing, we have 



As yet, there is no limitation on the value of Sa ; we now proceed to a limit 
by making a + Ba approach to coincidence with a, viz., by making Ba 
ultimately vanish. Taking moduli of each of the members of the last 
equation, we have 



(a) _ i f J(* 

2in j (z - o 



_ + . 



(z a) 2 (z a Ba) 



27T 



dz 



Let the greatest modulus of -. ~ =r. for points z along the 

(j a) 2 (z a Ba) 

boundary be M, which is a finite quantity on account of the conditions 
applying to f(z) and the fact that the points a and a + Ba are not 
infinitesimally near the boundary. Then, by 15, 



t 



dz 



0-a) 2 (z-a-Ba) 

<MS, 

where 8 is the whole length of the boundary, a finite quantity. Hence 

1 f f(z} , , |8a| 



dz 



c 

ITT 



21.] HOLOMORPHIC FUNCTIONS 31 

When we proceed to the limit in which Sa vanishes, we have Ba = 
and |o-| = 0, ultimately; hence the modulus on the left-hand side ultimately 
vanishes and therefore the quantity to which that modulus belongs is itself 
zero, that is, 



, 

(z of 

so that / (a) = -. !/-^~ n dz. 

ZTTI )(z- of 

This theorem evidently corresponds in complex variables to the well-known 
theorem of differentiation with respect to a constant under the integral 
sign when all the quantities concerned are real. 

Proceeding in the same way, we can prove that 

/ (a + &*)-/ (a) _ 2!_ f /(*) 
Ba ~2Trij(z-af 

where 6 is a small quantity which vanishes with Ba. Moreover the integral 
on the right-hand side is finite, for the subject of integration is everywhere 
finite along the path of integration which itself is of finite length. Hence, 
first, a small change in the independent variable leads to a change of the 
same order of small quantities in the value of the function f (a), which 
shews that f (a) is a continuous function. Secondly, denoting 

&*) -/(a) 



by &/ (), we have the limiting value of - * - equal to the integral on 

the right-hand side when Sa vanishes, that is, the derivative of f (a) has 
a value independent of the form of 8a and therefore / (a) is monogenic. 
Denoting this derivative by /"(a), we have 



J (z a) 3 

Thirdly, the function f (a) is uniform ; for it is the limit of the value 
of - -- x-- J-\J and both /(a) and /(a + Sa) are uniform. Lastly, it 

is finite; for (S 15) it is the value of the integral - . l.^^dz, in which 

2?n J (z af 

the length of the path is finite and the subject of integration is finite at 
every point of the path. 

Hence f (a) is continuous, monogenic, uniform, and finite throughout 
the whole of the region in which f (z) has these properties: it is a 
holomorphic function. Hence : 

When a function is holomorphic in any region of the plane bounded 



32 PROPERTIES OF [21. 

by a simple curve, its derivative is also holomorphic within that region. And, 
by repeated application of this theorem : 

When a function is holomorphic in any region of the plane bounded 
by a simple curve, it has an unlimited number of successive derivatives each 
of which is- holomorphic within the region. 

All these properties have been shewn to depend simply upon the holo 
morphic character of the fundamental function ; but the inferences relating 
to the derivatives have been proved only for points within the region and 
not for points on the boundary. If the foregoing methods be used to prove 
them for points on the boundary, they require that a consecutive point shall 
be taken in any direction ; in the absence of knowledge about the fundamental 
function for points outside (even though just outside) no inferences can be 
justifiably drawn. 

An illustration of this statement is furnished by the hypergeometric series 
which, together with all its derivatives, is holomorphic within a circle of 
radius unity and centre the origin ; and the series converges unconditionally 
everywhere on the circumference, provided 7 > a. + /3. But the corresponding 
condition for convergence on the circumference ceases to be satisfied for some 
one of the derivatives and for all which succeed it : as such functions do not 
then converge unconditionally, the circumference of the circle must be 
excluded from the region within which the derivatives are holomorphic. 

22. Expressions for the first and the second derivatives have been 
obtained. 

By a process similar to that which gives the value of f (a), the derivative 
of order n is obtainable in the form 

n f f (z\ 

/<) (a) = . I, dz, 
J w 2wt J (z - a) n+l 

the integral being taken round the whole boundary of the region or round any 
curves which arise from deformation of the boundary, provided that no point 
of the curves in the final or any intermediate form is indefinitely near to a. 

In the case when the curve of integration is a circle, no point of which 
circle may lie outside the boundary of the region, we have a modified form 
fcr /* (> 

For points along the circumference of the circle with centre a and radius 

r, let 

z a = re ei , 

dz 

so that as before = idO : 

z a 

then and 2?r being taken as the limits of 0, we have 



22.] HOLOMORPHIC FUNCTIONS 33 

Let M be the greatest value of the modulus of f (z) for points on the 
circumference (or, as it may be convenient to consider, of points on or within 
the circumference) : then 



\f (n) (a)\<~ e- nei \\f(a 

i / \ / 1 ^ 27ry w * - 

nl 

< 



M 



Now, let there be a function < (s) defined by the equation 

M 



a 



which can evidently be expanded in a series of ascending powers of z a 
that converges within the circle. The series is 



- 

[d n d> (z)~\ , M 

Hence !LJ =n \ 

[ d* ] z = a *> 

so that, if the value of the nth derivative of $(z), when z = a, be denoted 
by << n > (a), we have 

|/(a)| p>(a). 

These results can be extended to functions of more than one variable : 
the proof is similar to the foregoing proof. When the variables are two, 
say z and z , the results may be stated as follows : 

^ For all points z within a given simple curve in the ^-plane and all 
points / within a given simple curve G in the /-plane, let / (z, z) be a 
holomorphic function; then, if a be any point within C and a any point 
within G , 

^ n+n J (a, a ) 



J (z - a) n+1 (z aTf 

where n and ri are any integers and the integral is taken positively round the 
two curves G and G . 

If M be the greatest value of \f (z, z } for points z and z within their 
respective regions when the curves G and G are circles of radii r, r and 
centres a, a , then 

d n+n f(a, a ) M 

~3aW <w!/i! r v^5 

F. 



34 HOLOMORPHIC FUNCTIONS [ 22 - 

M 
and if $(?>*) 



d n+n f(a,a ) 



d n+n (j> (z, z ) 



then dada 

when z = a and z = a in the derivative of < (z, z). 

23. All the integrals of meromorphic functions that have been considered 
have been taken along complete curves : it is necessary to refer to integrals 
along curves which are lines only from one point to another. A single 
illustration will suffice at present. 

Consider the integral f -t-^-dz; the function / is 

J H z a 

supposed holomorphic in the given region, and z and z are 
any two points in that region. Let some curves joining z 
to z be drawn as in the figure (fig. 7). 

~ , * 2 o 

is holomorphic over the whole area en- Fig> 7 




z a 



closed by z^zSz : and therefore we have ^ = when taken round the 

boundary of that area. Hence as in the earlier case we have 



z a Jz z a 
The point a lies within the area enclosed by z yz^z , and the function 

is holomorphic, except in the immediate vicinity of z = a ; hence 

r f ( v\ 

I - dz = 2Trif(a), 

J z a 

the integral on the left-hand side being taken round Z yzj3z . Hence 



z a 



Denoting ^-by g(z), the function g (z) has one pole a in the region 

"~ CL 

considered. 

The preceding results are connected only with the simplest form of 
meromorphic functions; other simple results can be derived by means of the 
other theorems proved in 1721. Those which have been obtained are 
sufficient however to shew that : The integral of a meromorphic function 
fg(z)dz from one point to another of the region of the function is not in 
general a uniform function. The value of the integral is not altered by 
any deformation of the path which does not meet or cross a pole of the 
function; but the value is altered when the path of integration is so 



23.] GENERAL PROPOSITIONS IN INTEGRATION 35 

deformed as to pass over one or more poles. Therefore it is necessary to 
specify the path of integration when the subject of integration is a mero- 
morphic function ; only partial deformations of the path of integration are 
possible without modifying the value of the integral. 

24. The following additional propositions* are deduced from limiting 
cases of integration round complete curves. In the first, the curve becomes 
indefinitely small ; in the second, it becomes infinitely large. And in neither, 
are the properties of the functions to be integrated limited as in the pre 
ceding propositions, so that the results are of wider application. 

I. If f(z) be a function which, whatever be its character at a, has no 
infinities and no branch-points in the immediate vicinity of a, the value of 
ff(z)dz taken round a small circle with its centre at a tends towards zero 
when the circle diminishes in magnitude so as ultimately to be merely the 
point a, provided that, as z a diminishes indefinitely, the limit of (z a)f(z) 
tend uniformly to zero. 

Along the small circle, initially taken to be of radius r, let 

*-a-fl*i * 

dz 

so that = idO, 

z a 

and therefore Sf( z ) dz = i\ (z a)f(z) d6. 

Jo 

Hence \ff(z)dz\ = I *" (z - a)f(z) d0 

Jo 

<r\(z-a)f(z)\de 
Jo 

rzn 

< Md0 
Jo 



where M is the greatest value of M, the modulus of (z - a)f(z), for points 
on the circumference. Since (z - a)f(z) tends uniformly to the limit zero as 
| z -a diminishes indefinitely, \jf(z) dz\ is ultimately zero. Hence the integral 
itself jf(z)dz is zero, under the assigned conditions. 

Note. If the integral be extended over only part of the circumference of 
the circle, it is easy to see that, under the conditions of the proposition, 
the value offf(z)de still tends towards zero. 

COROLLARY. If (z-a)f(z) tend uniformly to a limit k as \z-a\ 
diminishes indefinitely, the value of ff(z)dz taken round a small circle centre 
a tends towards 27rik in the limit. 

* The form of the first two propositions, which is adopted here, is due to Jordan, Cours 
d Analyse, t. ii, 285, 286. 

32 



36 GENERAL PROPOSITIONS [24. 

Thus the value of [- dz j, taken round a very small circle centre , where a is 

~ d * / 2 V 

not the origin, is zero : the value of f - - - -, round the same circle is -. ( - \ . 

J (a z) (a-M) 

Neither the theorem nor the corollary will apply to a function, such as sn -^ 

which has the point a for an essential singularity: the value of (z-a)sn^ ^, as 
\z-a\ diminishes indefinitely, does not tend ( 13) to a uniform limit. As a matter of 

fact the function sn has an infinite number of poles in the immediate vicinity of a 
z- a 

as the limit za, is being reached. 

II. Whatever be the character of a function f (z} for infinitely large values 
ofz, the value ofjf(z) dz, taken round a circle with the origin for centre, tends 
towards zero as the circle becomes infinitely large, provided that, as \z\ 
increases indefinitely, the limit of zf(z) tend uniformly to zero. 

Along a circle, centre the origin and radius R, we have z =Ee ei , so that 

dz . ja 
- = idd, 

z 

r-2ir 

and therefore // 0) dz = i zf(z) d6. 

Jo 

Hence I //(*)<&! = * 

<T zf(z)\dS 
Jo 

rzn 

< Mde 

Jo 



< 

where M is the greatest value of M, the modulus of zf(z) t for points on 
the circumference. When R increases indefinitely, the value of M is zero 
on the hypothesis in the proposition; hence \$f(*)d*\ is ultimately zero. 
Therefore the value of ff(z) dz tends towards zero, under the assigned con 
ditions. 

Note. If the integral be extended along only a portion of the circumfer 
ence, the value of jf(z}dz still tends towards zero. 

COROLLARY. // zf(z) tend uniformly to a limit k as \z . increases 
indefinitely, the value of jf(z) dz, taken round a very large circle, centre the 
origin, tends towards %7rik. 

Thus the value of J(l -z n }~^dz round an infinitely large circle, centre the origin, is zero 
if n > 2, and is 2ir if = 2. 

III. If all the infinities and the branch-points of a function lie in a finite 
region of the z-plane, then the value of jf(z) dz round any simple curve, which 



24.] IN INTEGRATION 37 

includes all those points, is zero, provided the value of zf(z\ as \z\ increases 
indefinitely, tends uniformly to zero. 

The simple curve can be deformed continuously into the infinite circle 
of the preceding proposition, without passing over any infinity or any 
branch- point ; hence, if we assume that the function exists all over the plane, 
the value of jf(z) dz is, by Cor. I. of 19, equal to the value of the integral 
round the infinite circle, that is, by the preceding proposition, to zero. 

Another method of stating the proof of the theorem is to consider 
the corresponding simple curve on Neumann s sphere ( 4). The surface 
of the sphere is divided into two portions by the curve*: in one portion lie 
all the singularities and the branch-points, and in the other portion there is 
no critical point whatever. Hence in this second portion the function is holo- 
morphic ; since the area is bounded by the curve we see that, on passing back 
to the plane, the excluded area is one over which the function is holomorphic. 
Hence, by 19, the integral round the curve is equal to the integral round 
an infinite circle having its centre at the origin and is therefore zero, as 
before. 

COROLLARY. If, under the same circumstances, the value of zf(z}, as 
\z increases indefinitely, tend uniformly to k, then the value of $f(z)dz round 
the simple curve is 



Thus the value of I r along any simple curve which encloses the two points 
J (a 2 - z 2 )* 

a and - a is 2ir ; the value of 

dz 



{(!-") (!-*%)}* 

round any simple curve enclosing the four points 1, -1, T , -7, is zero, k being a non- 

1C K 

vanishing constant ; and the value of J(l z 2n )~*dz, taken round a circle, centre the origin 
and radius greater than unity, is zero when n is an integer greater than 1. 

/dz 
~ ~ 771 

K*-i) (*-)(*-)}* 

round any circle, which has the origin for centre and includes the three distinct points 
lt e 2 , e 3 , is not zero. The subject of integration has 2 = 00 for a branch-point, so that the 
condition in the proposition is not satisfied ; and the reason that the result is no longer 
valid is that the deformation into an infinite circle, as described in Cor. I. of 19, 
is not possible because the infinite circle would meet the branch-point at infinity. 

25. The further consideration of integrals of functions, that do not possess 
the character of uniformity over the whole area included by the curve of in 
tegration, will be deferred until Chap. ix. Some examples of the theorems 
proved in the present chapter will now be given. 

* The fact that a single path of integration is the boundary of two portions of the surface 
of the sphere, within which the function may have different characteristic properties, will be 
used hereafter ( 104) to obtain a relation between the two integrals that arise according as the 
path is deformed within one portion or within the other. 



38 EXAMPLES IN [25. 

Ex. 1. It is sufficient merely to mention the indefinite integrals (that is, integrals from 
an arbitrary point to a point z} of rational, integral, algebraical functions. After the 
preceding explanations it is evident that they follow the same laws as integrals of similar 
functions of real variables. 

/dz 
, ^ , taken round a simple curve. 

When n is 0, the value of the integral is zero if the curve do not include the point a, 
and it is Ziri if the curve include the point a. 

When n is a positive integer, the value of the integral is zero if the curve do not 
include the point a (by 17), and the value of the integral is still zero if the curve do 
include the point a (by 22, for the function f(z) of the text is 1 and all its derivatives 
are zero). Hence the value of the integral round any curve, which does not pass through 
a, is zero. 

We can now at once deduce, by 20, the result that, if a holomorphic function be 
constant along any simple closed curve within its region, it is constant over the whole 
area within the curve. For let t be any point within the curve, z any point on it, and C 
the constant value of the function for all the points z ; then 



B 



mn 

2 t 

the integral being taken round the curve, so that 

<&M- t dz 

= C 
by the above result, since the point t lies within the curve. 

Ex. 3. Consider the integral \e~^dz. 

In any finite part of the plane, the function e~ 02 is holomorphic; therefore ( 17) the 
integral round the boundary of a rectangle 
(fig. 8), bounded by the lines x= a, y = 0, 
y=b, is zero : and this boundary can be 
extended, provided the deformation remain 
in the region where the function is holo 
morphic. Now as a tends towards infinity, 
the modulus of e~ z \ being e~ x2 + y2 , tends 
towards zero when y remains finite ; and 
therefore the preceding rectangle can be Fig. 8. 

extended towards infinity in the direction of the axis of x, the side b of the rectangle 
remaining unaltered. 

Along A A, we have z=x : so that the value of the integral along the part A A of the 

fa 

boundary is I e~ x dx. 

J -a 

Along AB, we have z = a + iy, so that the value of the integral along the part AB 

f* 

is i I e~( a + iy rdy. 

Jo 
Along BB , we have z = x + ib, so that the value of the integral along the part BB 

f a 
is I e-( x + l Vdx. 

J a 

Along B A , we have z=-a + iy, so that the value of the integral along the part 

B A 1 is i (V(- H 
J t, 



25.] INTEGRATION 39 

/ft 
The second of these portions of the integral is e~ a<i . . I tP~*** t efy, which is easily seen 

J o 
to be zero when the (real) quantity a is infinite. 

Similarly the fourth of these portions is zero. 

Hence as the complete integral is zero, we have, on passing to the limit, 

I e~^dx+\ e-^ 2ibx + b 2 da;=0, 

J -<*> J oo 

whence e 62 I e~ *-***&?=* I 

J -oo J -<* 

/oo 
e 3 ^ (cos 2bxi sin 

and therefore, on equating real parts, we obtain the well-known result 



/ 

J -Q 



This is only one of numerous examples* in which the theorems in the text can be 
applied to obtain the values of definite integrals with real limits and real variables. 

r z n-i 
Ex. 4. Consider the integral I - --- dz. where n is a real positive quantity less than 

J 1+z 
unity. 

The only infinities of the subject of integration are the origin and the point - 1 ; 
the branch-points are the origin and 2=00. Everywhere else in the plane the function 
behaves like a holomorphic function ; and, therefore, when we take any simple closed 
curve enclosing neither the origin nor the point 1, the integral of the function round 
that curve is zero. 

We shall assume that the curve lies on the positive side of the axis of x and that it 
is made up of : 

(i) a semicircle (7 3 (fig. 9), centre the origin and radius R which is made to increase 
indefinitely : 




Fig. 9. 



(ii) two semicircles, c t and c 2 , with their centres at and 1 respectively, and with 

radii r and /, which ultimately are made infinitesimally small : 
(iii) the diameter of (7 3 along the axis of x excepting those ultimately infinitesimal 

portions which are the diameters of c x and of c 2 . 

The subject of integration is uniform within the area thus enclosed although it 
is not uniform over the whole plane. We shall take that value of z n ~ l which has its 
argument equal to (n 1) 6, where 6 is the argument of z. 

* See Briot and Bouquet, Theorie des fonctions elliptiques, (2nd ed.), pp. 141 et sqq., from 
which examples 3 and 4 are taken. 



40 EXAMPLES IN [25. 

The integral round the boundary is made up of four parts. 

0H 1 

(a) The integral round (7 3 . The value of z . , as z \ increases indefinitely, tends 

uniformly to the limit zero ; hence, as the radius of the semicircle is increased indefinitely, 
the integral round (7 3 vanishes ( 24, n., Note). 

^n 1 

(b) The integral round c v The value of z . , as | z \ diminishes indefinitely, 

1 -\-z 

tends uniformly to the limit zero ; hence as the radius of the semicircle is diminished 
indefinitely, the integral round c v vanishes ( 24, I., Note}. 

z n-l 

(c) The integral round c 2 . The value of (1 + 2) , as |1+2| diminishes indefinitely 

A ~r z 

for points in the area, tends uniformly to the limit ( I)"" 1 , i.e., to the limit g( M ~ 1 ) . 
Hence this part of the integral is 



being taken in the direction indicated by the arrow round c 2) the infinitesimal semicircle. 
Evidently -- =id6 and the limits are TT to 0, so that this part of the whole integral is 

idd 



(d) The integral along the axis of x. The parts at 1 and at which form the 
diameters of the small semicircles are to be omitted ; so that the value is 



-l+r J r 
This is what Cauchy calls the principal value* of the integral 

/" /yH ~ 1 

/ * 7 

I dx. 

Since the whole integral is zero, we have 

ine niri + I Y dx = 0. 

Let P = I ^ dx, P = I dx, 

and I dx, 

J o 1-a? 

principal values being taken in each case. Then, taking account of the arguments, we have 



Since iwe nvi + P + 1* = 0, 

we have P - e H7ri Q = - ine nni , 

* Williamson s Integral Calculus, % 104. 



25.] INTEGRATION 41 

so that P Q cos nn ir sin nn, Q sin ntr = TT cos nir. 

Hence I ; dxP = ir cosec TOTT, 

jo 1+a? 

dxQ = ir cot %TT. 



. 5. In the same way it may be proved that 



. 

where n is an integer, a is positive and o> is e* 2 " . 

Jik 6. By considering the integral Je- 2 ^- 1 ^ round the contour of the sector of a 
circle of radius r, bounded by the radii 0=0, 6=a, where a is less than |TT and n is positive, 
it may be proved that 



1 -; 



{r ; 
on proceeding to the limit when r is made infinite. (Briot and Bouquet.) 

Ex. 7. Consider the integral I ~~^, where n is an integer. The subject of integration 

is meromorphic ; it has for its poles (each of which is simple) the n points o> r for r=0, 
1, ..., n-l, where a is a primitive nth root of unity ; and it has no other infinities and no 

branch -points. Moreover the value of -, as \z\ increases indefinitely, tends uniformly 

to the limit zero ; hence ( 24, in.) the value of the integral, taken round a circle centre 
the origin and radius > 1, is zero. 

This result can be derived by means of Corollary II. in 19. Surround each of the 
poles with an infinitesimal circle having the pole for centre ; then the integral round 
the circle of radius > 1 is equal to the sum of the values of the integral round the 
infinitesimal circles. The value round the circle having r for its centre is, by 20, 



2rri( limit of " , when z = u> r } 
\ z - L J 



Hence the integral round the large circle 



2 

n r=n 



= 0. 



Ex. 8. Hitherto, in all the examples considered, the poles that have occurred have 
been simple : but the results proved in 21 enable us to obtain the integrals of 
functions which have multiple poles within an area. As an example, consider the 

integral / ( 1+g2 )n + i roun d any curve which includes the point i but not the point - i, these 
points being the two poles of the subject of integration, each of multiplicity n + l. 



42 EXAMPLES IN INTEGRATION [25. 

We have seen that /" (a) = j^ J ^_ a ^ n + i * 

where /() is holomorphic throughout the region bounded by the curve round which the 
integral is taken. 

In the present case a is i, and f(z) = . . n ^\ , s that 

2 ! (-l) n 



u. < /*n - 2- 

and therefore /" (*J = ^j (2i)*-n "~ ~ wT 

Hence we have "*** 



In the case of the integral of a function round a simple curve which contains several 
of its poles we first ( 20) resolve the integral into the sum of the integrals round simple 
curves each containing only one of the points, and then determine each of the latter 
integrals as above. 

Another method that is sometimes possible makes use of the expression of the uniform 
function in partial fractions. After Ex. 2, we need retain only those fractions which are of 

the form : the integral of such a fraction is ZniA, and the value of the whole integral 

z-a 

is therefore tor&A. It is thus sufficient to obtain the coefficients of the inverse first powers 
which arise when the function is expressed in partial fractions corresponding to each pole. 
Such a coefficient A, the coefficient of -j in the expansion of the function, is called by 

Z (Jj 

Cauchy the residue of the function relative to the point. 
For example, 



so that the residues relative to the points -1, -o>, -to 2 are f, , | 2 respectively. 
Hence if we take a semicircle, of radius > 1 and centre the origin with its diameter 
along the axis of y, so as to lie on the positive side of the axis of y, the area between the 
semi-circumference and the diameter includes the two points - and - 2 ; and therefore 

the value of 

dz 



taken along the semi-circumference and the diameter, is 

&*&+!?); 

i.e., the value is - *ni. 



CHAPTER III. 

EXPANSION OF FUNCTIONS IN SERIES OF POWERS. 

26. WE are now in a position to obtain the two fundamental theorems 
relating to the expansion of functions in series of powers of the variable : 
they are due to Cauchy and Laurent respectively. 

Cauchy s theorem is as follows*: 

When a function is holomorphic over the area of a circle of centre a, it can 
be expanded as a series of positive integral powers of z-a converging for all 
points within the circle. 

Let z be any point within the circle; describe a concentric circle of 
radius r such that 

\z-a\ = p <r<R, ^ ^i. 

where R is the radius of the given circle. If t 
denote a current point on the circumference of the 
new circle, we have 



dt 



t a z a 
t a 




Fif?. 10. 



the integral extending along the whole circumference of radius r. Now 



z-a 



t-a 



z -a n+l 



z a 



a 



ta 



so that, by 14 (III.), we have 



J_ f 
27ri] 



f(t) 



t-z\t-a 



dt. 



* Exercices d Analyse et de Physique Mathe matiqne, t. ii, pp. 50 et seq. ; the memoir was first 
made public at Turin in 1832. 






44 



CAUCHY S THEOREM ON THE [26. 

Now /() is holomorphic over the whole area of the circle ; hence, if t be 
not actually on the boundary of the region ( 21, 22), a condition secured by 
the hypothesis r < R, we have 



and therefore 

(z- a )n (z-a) n+l 

" 



Let the last term be denoted by L. Since z a =p and \t-a\ = r, 
it is at once evident that \t-z\^r-p. Let M be the greatest value of 
|/(0| for points along the circle of radius r ; then M must be finite, owing to 
the initial hypothesis relating tof(z). Taking 

f n TP 6i 

v W/ ~~ I C* 

so that dt = i(t- a) d6, 

P+> t*m de 

we have \L\ = - f 

i 

^P 



Jlf 



r n (i p) 

Jffl-: C 



\r) 
Now r was chosen to be greater than p ; hence as n becomes infinitely 

large, we have W infinitesimally small. Also If (1 ?l is finite. 
\r/ V f/ 

Hence as ?i increases indefinitely, the limit of |i|, necessarily not negative, 
is infinitesimally small and therefore, in the same case, L tends towards 
zero. 

It thus appears, exactly as in 15 (V.), that, when n is made to increase 
without limit, the difference between the quantity f(z) and the first n + 1 
terms of the series is ultimately zero ; hence the series is a converging series 
having f(z) as the limit of the sum, so that 



which proves the proposition under the assigned conditions. It is the form 
of Taylor s expansion for complex variables. 

Note. The series on the right-hand side is frequently denoted by 
P(z a), where P is a general symbol for a converging series of positive 
integral powers of z a: it is also sometimes* denoted by P(z\a). Con- 

* Weierstrass, Abh. am der Functionenlehre, p. 1. 



26.] EXPANSION OF A FUNCTION 45 

formably with this notation, a series of negative integral powers of z a 

would be denoted by P I - ) ; a series of negative integral powers of z 

\z a/ 

either by P (-) or by P(^|oo), the latter implying a series proceeding in 
\zj 

positive integral powers of a quantity which vanishes when z is infinite, 
i.e., in positive integral powers of . 

Z 

If, however, the circle can be made of infinitely great radius so that the 
function f(z) is holomorphic over the finite part of the plane, the equivalent 
series is denoted by G(z a) and it converges over the whole plane. 
Conformably with this notation, a series of negative integral powers of z - a 

which converges over the whole plane is denoted by G I - - j . 

27. The following remarks on the proof and on inferences from it should 
be noticed. 

(i) In order that - - may be expanded in the required form, the 
t z 

point z must be taken actually within the area of the circle of radius R ; 
and therefore the convergence of the series P (z a) is not established for 
points on the circumference. 

(ii) The coefficients of the powers of z a in the series are the 
values of the function and its derivatives at the centre of the circle ; and the 
character of the derivatives is sufficiently ensured ( 21) by the holomorphic 
character of the function for all points within the region. It therefore 
follows that, if a function be holomorphic within a region bounded by a 
circle of centre a, its expansion in a series of ascending powers of z a 
converging for all points within the circle depends only upon the values of 
the function and its derivatives at the centre. 

But instead of having the values of the function and of all its derivatives 
at the centre of the circle, it will suffice to have the values of the holomorphic 
function itself over any small region at a or along any small line through 
a, the region or the line not being infinitesimal. The values of the 
derivatives at a can be found in either case ; for / (b) is the limit of 
{f(b + 86) f(b)}/8b, so that the value of the first derivative can be found 
for any point in the region or on the line, as the case may be ; and so for all 
the derivatives in succession. 

(iii) The form of Maclaurin s series for complex variables is at once 
derivable by supposing the centre of the circle at the origin. We then 
infer that, if a function be kolomorphic over a circle, centre tJie origin, it can be 



46 DARBOUX S EXPRESSION [27. 

represented in the form of a series of ascending, positive, integral powers of the 
variable given by 



where the coefficients of the various powers of z are the values of the derivatives 
of f(z) at the origin, and the series converges for all points within the circle. 

Thus, the function e z is holomorphic over the finite part of the plane ; 
therefore its expansion is of the form G (z). The function log (1 4- z) has a 
singularity at 1 ; hence within a circle, centre the origin and radius unity, 
it can be expanded in the form of an ascending series of positive integral 
powers of z, it being convenient to choose that one of the values of the 
function which is zero at the origin. Again, tan" 1 .? 2 has singularities at the 
four points z 4 = I, which all lie on the circumference; choosing the value at 
the origin which is zero there, we have a similar expansion in a series, con 
verging for points within the circle. 

Similarly for the function (1 +z) n , which has 1 for a singularity. 

(iv) Darboux s method* of derivation of the expansion of f (z) in 
positive powers of z a depends upon the expression, obtained in 15 (IV.), 
for the value of an integral. When applied to the general term 

1 Uz-a\ n+i s,.. ,, 
f(t)dt, 



= L say, it gives L = \r fe^J /(f), 

where is some point on the circumference of the circle of radius r, and X is 

2 ~ fl 

a complex quantity of modulus not greater than unity. The modulus of ^ 

b ~~ a 

is less than a quantity which is less than unity ; the terms of the series of 
moduli are therefore less than the terms of a converging geometric progres 
sion, so that they form a converging series; the limit of \L\, and therefore 
of L, can, with indefinite increase of n, be made zero and Taylor s expansion 
can be derived as before. 

00 

Ex. 1. Prove that the arithmetic mean of all values of z~ n 2 a v z v , for points lying along 

v = 

a circle |z| = r entirely contained in the region of continuity, is a n . (Rouche, Gutzmer.) 
Prove also that the arithmetic mean of the squares of the moduli of all values of 

00 

2 a v z v , for points lying along a circle z\ = r entirely contained in the region of continuity, 

x = 

is equal to the sum of the squares of the moduli of the terms of the series for a point on 
the circle. (Gutzmer.) 

00 

Ex. 2. Prove that the function 2 a n z n *, 

M = 

is finite and continuous, as well as all its derivatives, within and on the boundary of the 
circle |0| = 1, provided a < 1. (Fredholm.) 

* Liouville, 3 dmc Ser., t. ii, (1876), pp. 291312. 



28.] 



LAURENT S EXPANSION OF A FUNCTION 



47 



28. Laurent s theorem is as follows*: 

A function, which is holomorphic in a part of the plane bounded by two 
concentric circles with centre a and finite radii, can be expanded in the form 
of a double series of integral powers, positive and negative, of z a, the series 
converging uniformly and unconditionally in the part of the plane between the 
circles. 

Let z be any point within the region bounded by the two circles of radii 
R and R; describe two concentric circles of 
radii r arid r such that 

R>r> z-a >r > R. 

Denoting by t and by s current points on the 
circumference of the outer and of the inner 
circles respectively, and considering the space 
which lies between them and includes the point 
z, we have, by 20, 

/w-oL 




: Z ZTTlJs 2"~ Fig. 11. 

a negative sign being prefixed to the second integral because the direction 
indicated in the figure is the negative direction for the description of the 
inner circle regarded as a portion of the boundary. 

Now we have 

fz a" 
t a _ z 

t 



a Iz a\* 
a \t-aj 



z- a. 

+ I . - + 
a. 



1 - 



z a 



t a 

this expansion being adopted with a view to an infinite converging series, 
z a 



because 



t a 



is less than unity for all points t; and hence, by 15, 



_ n\n+l 



dt. 



z \t a/ 

Now each of the integrals, which are the respective coefficients of powers of 
z a, is finite, because the subject of integration is everywhere finite along 
the circle of finite radius, by 15 (IV.). Let the value of 

^r* % - - 

be 2iriu r : the quantity u r is not necessarily equal to / (a) -r- r I, because no 
* Comptes Rendus, t. xvii, (1843), p. 939. 






48 LAURENT S EXPANSION OF [28. 

knowledge of the function or of its derivatives is given for a point within 
the innermost circle of radius R . Thus 

_L f/2) dt = u + (z - a) u 1 + (z- a) 2 w 2 + +(z- a) n u n 

2w J t z 

1 [f (t) (z a\ n+1 -, 



- z \t a 

The modulus of the last term is less than 

M 



where p is z-a and If is the greatest value of \f(t)\ for points along the 
circle. Because p < r, this quantity diminishes to zero with indefinite in 
crease of n ; and therefore the modulus of the expression 



v % 

becomes indefinitely small with increase of n. The quantity itself therefore 
vanishes in the same limiting circumstance ; and hence 

1 . [fl&dt = u + (z-<i)u 1 + ...... +(z-a) m u m + ...... , 

2-7TI J t Z 

so that the first of the integrals is equal to a series of positive powers. This 
series converges uniformly and unconditionally within the outer circle, for 
the modulus of the (m + l) th term is less than 



which is the (m + l) th term of a converging series*. 

As in 27, the equivalence of the integral and the series can be affirmed 
only for points which lie within the outermost circle of radius R. 

Again, we have 

fs - a\ n+1 



z-a _ s-a fs - a\ n (z-a) 



s-z z-a \z-a 

z a 

this expansion being adopted with a view to an infinite converging series, 



because 



s a 



z a 



is less than unity. Hence 



1 [/s-a\ 
. If - - 
2?rt J \z-aj 



- n+1 f(s) 
J -~- 



, 

-ds. 
z s 



Chrystal, ii, 124. 



28.] A FUNCTION IN SERIES 49 

The modulus of the last term is less than 

M 




P 

where M is the greatest value of \f(s)\ for points along the circle of radius 
r . With indefinite increase of n, this modulus is ultimately zero ; and thus, 
by an argument similar to the one which was applied to the former integral, 
we have 



.. .. - .. 

ZTTI J s z z a (z a) 2 (z a) m 

where v m denotes the integral f(s a) m ~ l f (s) ds taken round the circle. 

As in the former case, the series is one which converges uniformly and 
unconditionally; and the equivalence of the integral and the series is valid 
for points z that lie without the innermost circle of radius R . 

The coefficients of the various negative powers of z a are of the form 

1 f /(*) d ( 1 ^ 
tori] __ 1_ (s-a) 

(s - a) m 
a form that suggests values of the derivatives of f (s) at the point given by 

- = 0, that is, at infinity. But the outermost circle is of finite radius ; 
s-a 

and no knowledge of the function at infinity, lying without the circle, is 
given, so that the coefficients of the negative powers may not be assumed 
to be the values of the derivatives at infinity, just as, in the former case, the 
coefficients u r could not be assumed to be the values of the derivatives at the 
common centres of the circles. 

Combining the expressions obtained for the two integrals, we have 
f(z) = u + (z a) u-i + (z a) 2 w 2 + ... 

+ (z- a)- 1 Vl + (z- a)~ 2 v a + .... 

Both parts of the double series converge uniformly and unconditionally for 
all points in the region between the two circles, though not necessarily for 
points on the boundary of the region. The whole series therefore converges 
for all those points : and we infer the theorem as enunciated. 

Conformably with the notation ( 26, note) adopted to represent Taylor s 
expansion, a function f(z) of the character required by Laurent s Theorem 
can be represented in the form 



the series P 1 converging within the outer circle and the series P 2 converging 
without the inner circle ; their sum converges for the ring-space between the 
circles. 

F. 4 



50 LAURENT S THEOREM [29. 

29. The coefficient u in the foregoing expansion is 

-1- f 9 dt 

torijt-a 

the integral being taken round the circle of radius r. We have 

dt =ide 



t a 
for points on the circle ; and therefore 

d0 



so that \u \<! de M t <M , 

J ZTT 

M being the greatest value of M t , the modulus of f(t), for points along the 
circle. If M be the greatest value of \f(z}\ for any point in the whole 
region in which f(z) is defined, so that M ^.M, then we have 

o 1 < M, 

that is, the modulus of the term independent of z a in the expansion of 
f(z) by Laurent s Theorem is less than the greatest value of \f(z) \ at points 
in the region in which it is defined. 

Again, (z-a)- m f(z) is a double series in positive and negative powers of 
z-a, the term independent of z -a being u m ; hence, by what has just been 
proved, u m \ is less than p~ m M, where p is z - a . But the coefficient u m 
does not involve z, and we can therefore choose a limit for any point z. The 
lowest limit will evidently be given by taking z on the outer circle of radius 
R, so that u m < MR~ m . Similarly for the coefficients v m ; and therefore we 
have the result : 

If f(z) be expanded as by Laurent s Theorem in the form 

OO 00 

u + 2 (z-a) m u m + 2 (z-aY^Vm, 

m = l m=l 

then \u m <MR~ m , \v m <MR m , 

where M is the greatest value of \f(z) at points within the region in which 

f(z) is defined, and R and R are the radii of the outer and the inner circles 

respectively. 

30. The following proposition is practically a corollary from Laurent s 
Theorem : 

When a function is holomorphic over all the plane which lies outside a 
circle of centre a, it can be expanded in the form of a series of negative integral 
powers of z a, the series converging uniformly and unconditionally everywhere 
in that part of the plane. 

It can be deduced as the limiting case of Laurent s Theorem when the 



30.] EXPANSION IN NEGATIVE POWERS 51 

radius of the outer circle is made infinite. We then take r infinitely large, 
and substitute for t by the relation 

t a = re ei , 

so that the first integral in the expression (a), p. 47, for/(^) is 

1 f 2 " d0 



t a 

Since the function is holomorphic over the whole of the plane which lies 
outside the assigned circle, f(t} cannot be infinite at the circle of radius r 
when that radius increases indefinitely. If it tend towards a (finite) limit k, 
which must be uniform owing to the hypothesis as to the functional character 
of f(z\ then, since the limit of (t z)/(t a) is unity, the preceding integral 
is equal to k. 

The second integral in the same expression (a), p. 47, for f(z) is un 
altered by the conditions of the present proposition ; hence we have 

f(z) = k + (z- a)~ l v l + (z- a)- 2 Vz + ..., 

the series converging uniformly and unconditionally without the circle, 
though it does not necessarily converge on the circumference. 

The series can be represented in the form 

1 



\z a/ 
conformably with the notation of 26. 

Of the three theorems in expansion which have been obtained, Cauchy s 
is the most definite, because the coefficients of the powers are explicitly 
obtained as values of the function and of its derivatives at an assigned point. 
In Laurent s theorem, the coefficients are not evaluated into simple expres 
sions ; and in the corollary frofti Laurent s theorem the coefficients are, as is 
easily proved, the values of the function and of its derivatives for infinite 
values of the variable. The essentially important feature of all the theorems 
is the expansibility of the function in series under assigned conditions. 

31. It was proved (21) that, when a function is holomorphic in any 
region of the plane bounded by a simple curve, it has an unlimited number 
of successive derivatives each of which is holomorphic in the region. Hence, 
by the preceding propositions, each such derivative can be expanded in 
converging series of integral powers, the series themselves being deducible 
by differentiation from the series which represents the function in the region. 

In particular, when the region is a finite circle of centre a, within which 
f(z) and consequently all the derivatives off(z) are expansible in converging 
series of positive integral powers of z a, the coefficients of the various 
powers of z a are save as to numerical factors the values of the 

42 



52 DEFINITION OF DOMAIN [31. 

derivatives at the centre of the circle. Hence it appears that, when a function 
is holomorphic over the area of a given circle, the values of the function and all 
its derivatives at any point z within the circle depend only upon the variable 
of the point and upon the values of the function and its derivatives at the 
centre. 

32. Some of the classes of points in a plane that usually arise in 
connection with uniform functions may now be considered. 

(i) A point a in the plane may be such that a function of the variable 
has a determinate finite value there, always independent of the path by 
which the variable reaches a ; the point a, is called an ordinary point* of the 
function. The function, supposed continuous in the vicinity of a, is con 
tinuous at a : and it is said to behave regularly in the vicinity of an ordinary 
point. 

Let such an ordinary point a be at a distance d, not infinitesimal, from 
the nearest of the singular points (if any) of the function ; and let a circle of 
centre a and radius just less than d be drawn. The part of the z-plane lying 
within this circle is calledf the domain of a ; and the function, holomorphic 
within this circle, is said to behave regularly (or to be regular) in the domain 
of a. From the preceding section, we infer that a function and its derivatives 
can be expanded in a converging series of positive integral powers of z a 
for all points z in the domain of a, an ordinary point of the function : and 
the coefficients in the series are the values of the function and its derivatives 
at a. 

The property possessed by the series that it contains only positive 
integral powers of z - a at once gives a test that is both necessary and 
sufficient to determine whether a point is an ordinary point. If the point a 
be ordinary, the limit of (z - a) f (z} necessarily is zero when z becomes equal 
to a. This necessary condition is also sufficient to ensure that the point is 
an ordinary point of the function / (z), supposed to be uniform ; for, since 
f(z) is holomorphic, the function (z-a)f(z) is also holomorphic and can be 
expanded in a series 

M -f w a (z d) + w 2 (? a) 2 + -, 

converging in the domain of a. The quantity u is zero, being the value 
of (z-a)f(z) at a and this vanishes by hypothesis; hence 

(z-a)f (z) = (z a) {MI + u 2 (z -a) +...}, 

shewing that / (z) is expressible as a series of positive integral powers of 
z a converging within the domain of a, or, in other words, that/(*) certainly 
has a for an ordinary point in consequence of the condition being satisfied. 

* Sometimes a regular point. 

t The German title is Umgebung, the French is domaine. 



32.] ESSENTIAL SINGULARITY 53 

(ii) A point a in the plane may be such that a function / (z) of the 
variable has a determinate infinite value there, always independent of the 
path by which the variable reaches a, the function behaving regularly for 

points in the vicinity of a ; then ^\ nas a determinate zero value there, so 

/ (?) 

that a is an ordinary point of --r-r . The point a is called a pole (12) or 
an accidental singularity* of the function. 

A test, necessary and sufficient to settle whether a point is an accidental 
singularity of a function will subsequently ( 42) be given. 

(iii) A point a in the plane may be such that y (2) has not a determinate 
value there, either finite or infinite, though the function is regular for all 

points in the vicinity of a that are not at merely infinitesimal distances. 

i 1 
Thus the origin is of this nature for the functions e z , sn - . 

Z 

Such a point is called-f* an essential singularity of the function. No 
hypothesis is postulated as to the character of the function for points 
at infinitesimal distances from the essential singularity, while the relation 
of the singularity to the function naturally depends upon this character at 
points near it. There may thus be various kinds of essential singularities 
all included under the foregoing definition ; their classification is effected 
through the consideration of the character of the function at points in their 
immediate vicinity. (See 88.) 

One sufficient test of discrimination between an accidental singularity 
and an essential singularity is furnished by the determinateness of the value 
at the point. If the reciprocal of the function have the point for an ordinary 
point, the point is an accidental singularity it is, indeed, a zero for the 
reciprocal. But when the point is an essential singularity, the value of the 
reciprocal of the function is not determinate there ; and then the reciprocal, 
as well as the function, has the point for an essential singularity. 

33. It may be remarked at once that there must be at least one 
infinite value among the values which a function can assume at an essential 
singularity. For if/ (z) cannot be infinite at a, then the limit of (z a)f (z) 
is zero when z = a, no matter what the non-infinite values of f (z) may be, 
that is, the limit is a determinate zero. The function (z a)f(z) is regular 
in the vicinity of a : hence by the foregoing test for an ordinary point, 
the point a is ordinary and the value of the uniform function f(z) is 

* Weierstrass, Abh. aus der Functionenlehre, p. 2, to whom the name is due, calls it ausser- 
wesentliche singuldre Stelle ; the term non-essential is suggested by Mr Cathcart, Harnack, p. 148. 
t Weierstrass, I.e., calls it wesentliche singulare Stelle. 



54 CONTINUATIONS OF A FUNCTION [33. 

determinate, contrary to hypothesis. Hence the function must have at least 
one infinite value at an essential singularity. 

Further, a uniform function must be capable of assuming any value C at 
an essential singularity. For an essential singularity of / (z) is also an 

essential singularity of / (z) G and therefore also of .. \_n The last 

function must have at least one infinite value among the values that it can 
assume at the point ; and, for this infinite value, we have / (z) C at the 
point, so that/(f) assumes the assigned value C at the essential singularity*. 

34. Let f(z) denote the function represented by a series of powers 
Pj (z a), the circle of convergence of which is the domain of the ordinary 
point a of the function. The region over which the function / (z) is holo- 
morphic may extend beyond the domain of a, although the circumference 
bounding that domain is the greatest of centre a that can be drawn within 
the region. The region evidently cannot extend beyond the domain of a in 
all directions. 

Take an ordinary point b in the domain of a. The value at b of the 
function /(V) is given by the series Pj (b a), and the values at b of all its 
derivatives are given by the derived series. All these series converge within 
the domain of a and they are therefore finite at b ; and their expressions 
involve the values at a of the function and its derivatives. 

Let the domain of b be formed. The domain of b may be included in 
that of a, and then its bounding circle will touch the bounding circle of the 
domain of a internally. If the domain of b be not entirely included in that 
of a, part of it will lie outside the domain of a ; but it cannot include the 
whole of the domain of a unless its bounding circumference touch that of the 
domain of a externally, for otherwise it would extend beyond a in all 
directions, a result inconsistent with the construction of the domain of a. 
Hence there must be points excluded from the domain of a which are also 
excluded from the domain of b. 

For all points z in the domain of b, the function can be represented by a 
series, say P 2 (2 b), the coefficients of which are the values at b of the 
function and its derivatives. Since these values are partially dependent 
upon the corresponding values at a, the series representing the function may 
be denoted by P 2 (z b, a). 

At a point z in the domain of b lying also in the domain of a, the two 
series P l (z a) and P 2 (z b, a) must furnish the same value for the 
function / (V) ; and therefore no new value is derived from the new series P 2 

* Weierstrass, I.e., pp. 5052; Durege, Elemente der Theorie der Funktionen, p. 119; Holder, 
Math. Ann., t. xx, (1882), pp. 138 143 ; Picard, " Memoire sur les fonctions entieres," Annahs de 
VEcole Norm. Sup., 2 me Ser., t. ix, (1880), pp. 145 166, which, in this regard, should be consulted 
in connection with the developments in Chapter V. See also 62. 



34.] OVER ITS REGION OF CONTINUITY 55 

which cannot be derived from the old series Pj. For all such points the new 
series is of no advantage ; and hence, if the domain of b be included in that 
of a, the construction of the series P 2 (z b, a) is superfluous. Hence in 
choosing the ordinary point b in the domain of a we choose a point, if 
possible, that will not have its domain included in that of a. 

At a point z in the domain of b, which does not lie in the domain of a, 
the series P 2 (z b, a) gives a value for f(z) which cannot be given by 
P l (z a). The new series P 2 then gives an additional representation of the 
function ; it is called* a continuation of the series which represents the function 
in the domain of a. The derivatives of P 2 give the values of f(z) for points 
in the domain of b. 

It thus appears that, if the whole of the domain of b be not included in 
that of a, the function can, by the series which is valid over the whole 
of the new domain, be continued into that part of the new domain excluded 
from the domain of a. 

Now take a point c within the region occupied by the combined domains 
of a and b ; and construct the domain of c. In the new domain, the function 
can be represented by a new series, say P 3 (z c), or, since the coefficients 
(being the values at c of the function and of its derivatives) involve the 
values at a and possibly also the values at b of the function and of its 
derivatives, the series representing the function may be denoted by 
P z (z c, a, b). Unless the domain of c include points, which are not 
included in the combined domains of a and b, the series P 3 does not give 
a value of the function which cannot be given by Pj or P 2 : we therefore 
choose c, if possible, so that its domain will include points not included in 
the earlier domains. At such points z in the domain of c as are excluded 
from the combined domains of a and 6, the series P 3 (z c, a, b) gives a value 
for f(z) which cannot be derived from P 1 or P 2 ; and thus the new series 
is a continuation of the earlier series. 

Proceeding in this manner by taking successive points and constructing 
their domains, we can reach all parts of the plane connected with one 
another where the function preserves its holomorphic character; their 
combined aggregate is called -f the region of continuity of the function. 
With each domain, constructed so as to include some portion of the region of 
continuity not included in the earlier domains, a series is associated, which is 
a continuation of the earlier series and, as such, gives a value of the function 
not deducible from those earlier series ; and all the associated series are 
ultimately derived from the first. 

* Biermann, Theorie der analytischen Functional, p. 170, which may be consulted in 
connection with the whole of 34; the German word is Fortsetzung. 
t Weierstrass, I.e., p. 1. 



56 DEFINITION OF ANALYTIC FUNCTION [34. 

Each of the continuations is called an Element of the function. The 
aggregate of all the distinct elements is called a monogenic analytic function : 
it is evidently the complete analytical expression of the function in its region 
of continuity. 

Let z be any point in the region of continuity, not necessarily in the 
circle of convergence of the initial element of the function; a value of the 
function at z can be obtained through the continuations of that initial 
element. In the formation of each new domain (and therefore of each new 
element) a certain amount of arbitrary choice is possible ; and there may, 
moreover, be different sets of domains which, taken together in a set, each 
lead to z from the initial point. When the analytic function is uniform, as 
before defined ( 12), the same value at z for the function is obtained, 
whatever be the set of domains. If there be two sets of elements, differently 
obtained, which give at z different values for the function, then the ana 
lytic function is multiform, as before defined ( 12) ; but not every change 
in a set of elements leads to a change in the value at z of a multiform 
function, and the analytic function is uniform within such a region of the 
plane as admits only equivalent changes of elements. 

The whole process is reversible when the function is uniform. We can 
pass back from any point to any earlier point by the use, if necessary, of 
intermediate points. Thus, if the point a in the foregoing explanation 
be not included in the domain of b (there supposed to contribute a continu 
ation of the first series), an intermediate point on a line, drawn in the 
region of continuity so as to join a and b, would be taken ; and so on, 
until a domain is formed which does include a. The continuation, associated 
with this domain, must give at a the proper value for the function and its 
derivatives, and therefore for the domain of a the original series P l (z a) 
will be obtained, that is, Pj (z a) can be deduced from P 2 (z b, a) the 
series in the domain of b. This result is general, so that any one of the 
continuations of a uniform function, represented by a power-series, can be 
derived from any other; and therefore the expression of such a function in 
its region of continuity is potentially given by one element, for all the 
distinct elements can be derived from any one element. 

35. It has been assumed that the property, characteristic of some of the 
functions adduced as examples, of possessing either accidental or essential 
singularities, is characteristic of all functions ; it will be proved ( 40) to hold 
for every uniform function which is not a mere constant. 

The singularities limit the region of continuity ; for each of the separate 
domains is, from its construction, limited by the nearest singularity, and the 
combined aggregate of the domains constitutes the region of continuity when 



35.] 



SCHWARZ S CONTINUATION 



57 



they form a continuous space*. Hence the complete boundary of the region 
of continuity is the aggregate of the singularities of the function-}-. 

It may happen that a function has no singularity except at infinity ; the 
region of continuity then extends over the whole finite part of the plane but 
it does not include the point at infinity. 

It follows from the foregoing explanations that, in order to know a 
uniform analytic function, it is necessary to know some element of the 
function, which has been shewn to be potentially sufficient for the derivation 
of the full expression of the function and for the construction of its region of 
continuity. 

36. The method of continuation of a function, which has just been 
described, is quite general ; there is one particular continuation, which is 
important in investigations on conformal representations. It is contained in 
the following proposition, due to SchwarzJ : 

If an analytic function w of z be defined only for a region 8 in the 
positive half of the z-plane and if continuous real values of w correspond to 
continuous real values of z, then w can be continued across the axis of real 
quantities. 

Consider a region 8", symmetrical with S relative to the axis of real 
quantities (fig. 12). Then a function is 
defined for the region S" by associating 
a value w , the conjugate of w, with z , 
the conjugate of z. 

Let the two regions be combined along 
the portion of the axis of ac which is their 
common boundary ; they then form a 
single region S + S". 

Consider the integrals 




Fig. 12. 



1 [ w j A ! [ w o 
o I i-dz and ^ -. / 
fcp/fjr-f 2w./,rt- 



taken round the boundaries of 8 and of 8" respectively. Since w is 

* Cases occur in which the region of continuity of a function is composed of isolated spaces, 
each continuous in itself, but not continuous into one another. The consideration of such cases 
will be dealt with briefly hereafter, and they are assumed excluded for the present : meanwhile, 
it is sufficient to note that each continuous space could be derived from an element belonging to 
some domain of that space and that a new element would be needed for a new space. 

t See Weierstrass, I.e., pp. 13 ; Mittag-Leffler, " Sur la representation analytique des fonctions 
monogenes uniformes d une variable independante," Acta Math., t. iv, (1884), pp. 1 et seq., 
especially pp. 1 8. 

Crelle, t. Ixx, (1869), pp. 106, 107, and Ges. Math. Abh., t. ii, pp. 6668. See also Darboux, 
Theorie generate des surfaces, t. i, 130. 



58 SCHWARZ S CONTINUATION [36. 

continuous over the whole area of 8 as well as along its boundary and 
likewise w relative to 8", it follows that, if the point f be in 8 , the value of 
the first integral is w (f ) and that of the second is zero ; while, if lie in 8", 
the value of the first integral is zero and that of the second is w (). Hence 
the sum of the two integrals represents a unique function of a point in either 
8 or 8". But the value of the first integral is 



M wdz J^ [ B w Q) dap 

I (" ~~ (f C\ I V> 

J ji 2 ZTriJ A x L, 



the first being taken along the curve EC. A and the second along the axis 
AxB ; and the value of the second integral is 

1 C A w (x)dx 1_ f * W dz 
2-Tri J B x ZTTI J A *o 
the first being taken along the axis Ex A and the second along the curve 

ADB. But 

w (ac) = w (x), 

because conjugate values w and w correspond to conjugate values of the 
argument by definition of W and because w (and therefore also w ) is real 
and continuous when the argument is real and continuous. Hence when the 
sum of the four integrals is taken, the two integrals corresponding to the 
two descriptions of the axis of x cancel and we have as the sum 

wdz 1 



A 



and this sum represents a unique function of a point in 8 + 8". These two 
integrals, taken together, are 

_L [ w dz 
2Tn]z-t 

taken round the whole contour of 8 + 8", where w is equal to w (f) in the 
positive half of the plane and to w (^) in the negative half. 

For all points in the whole region 8 + 8", this integral represents a 
single uniform, finite, continuous function of f; its value is w () in the 
positive half of the plane and is w (f) in the negative half; and therefore 
w () is the continuation into the negative half of the plane of the function, 
which is defined by w () for the positive half. 

For a point c on the axis of x, we have 

w (z) -w(c) = A(z-c) + B(z-cy>+C(z-cY + ...; 

and all the coefficients A, B, C,... are real. If, in addition, w be such a 
function of z that the inverse functional relation makes z a uniform 
analytic function of w, it is easy to see that A must not vanish, so that the 
functional relation may be expressed in the form 

w(z)w (c) = (z-c}P(z- c), 
where P (z c) does not vanish when z = c. 



CHAPTER IV. 

GENERAL PROPERTIES OF UNIFORM FUNCTIONS, PARTICULARLY OF THOSE 
WITHOUT ESSENTIAL SINGULARITIES. 

37. IN the derivation of the general properties of functions, which will be 
deduced in the present and the next three chapters from the results already 
obtained, it is to be supposed, in the absence of any express statement to 
other effect, that the functions are uniform, monogenic and, except at either 
accidental or essential singularities, continuous*. 

THEOREM I. A function, which is constant throughout any region of the 
plane not infinitesimal in area, or which is constant along any line not infini 
tesimal in length, is constant throughout its region of continuity. 

For the first part of the theorem, we take any point a in the region of the 
plane where the function is constant, and we draw a circle of centre a and 
of any radius, provided only that the circle remains within the region of 
continuity of the function. At any point z within this circle we have 

/<*) =/(a) + (z - a)f (a) + ( -, ~^ f" (a) + . . , 

a converging series the coefficients of which are the values of the function 
and its derivatives at a. But 

/X) = Limit of ^MZ/^), :. V, : 

which is zero because f(a + Ba) is the same constant as f(a) : so that the 
first derivative is zero at a. Similarly, all the derivatives can be shewn to 
be zero at a ; hence the above series after its first term is evanescent, 
and we have 

/(*)-/<), 

that is, the function preserves its constant value throughout its region of 
continuity. 

The second result follows in the same way, -when once the derivatives are 
proved zero. Since the function is monogenic, the value of the first and 

* It will be assumed, as in 35 (note, p. 57), that the region of continuity consists of a single 
space ; functions, with regions of continuity consisting of a number of separated spaces, will be 
discussed in Chap. VII. 



60 ZEROS OF A [37. 

of each of the successive derivatives will be obtained, if we make the 
differential element of the independent variable vanish along the line. 

Now, if a be a point on the line and a + 8a a consecutive point, we have 
f(a + So) = f(a) ; hence / (a) is zero. Similarly the first derivative at any 
other point on the line is zero. Therefore we have / (a + So) =f (a), for 
each has just been proved to be zero : hence /" (a) is zero ; and similarly the 
value of the second derivative at any other point on the line is zero. So on 
for all the derivatives : the value of each of them at a is zero. 

Using the same expansion as before and inserting again the zero values 
of all the derivatives at a, we find that 

/(*)=/(), 

so that under the assigned condition the function preserves its constant value 
throughout its region of continuity. 

It should be noted that, if in the first case the area be so infinitesimally 
small and in the second the line be so infinitesimally short that consecutive 
points cannot be taken, then the values at a of the derivatives cannot be 
proved to be zero and the theorem cannot then be inferred. 

COROLLARY I. If two functions have the same value over any area of 
their common region of continuity which is not infinitesimally small or along 
any line in that region which is not infinitesimally short, then they have the 
same values at all points in their common region of continuity. 

This is at once evident : for their difference is zero over that area or along 
that line and therefore, by the preceding theorem, their difference has a 
constant zero value, that is, the functions have the same values, everywhere 
in their common region of continuity. 

But two functions can have the same values at a succession of isolated 
points, without having the same values everywhere in their common region 
of continuity ; in such a case the theorem does not apply, the reason being 
that the fundamental condition of equality over a continuous area or along 
a continuous line is not satisfied. 

COROLLARY II. A function cannot be zero over any continuous area of its 
region of continuity which is not infinitesimal or along any line in that region 
which is not infinitesimally short without being zero everywhere in its region of 
continuity. 

This corollary is deduced in the same manner as that which precedes. 

If, then, there be a function which is evidently not zero everywhere, we 
conclude that its zeros are isolated points though such points may be multiple 
zeros. 

Further, in any finite area of the region of continuity of a function that is 
subject to variation, there can be at most only a finite number of its zeros, when 



37.] UNIFORM FUNCTION 61 

no point of the boundary of the area is infinitesimally near an essential 
singularity. For if there were an infinite number of such points in any 
such region, there must be a cluster in at least one area or a succession 
along at least one line, infinite in number and so close as to constitute a 
continuous area or a continuous line where the function is everywhere zero. 
This would require that the function should be zero everywhere in its region 
of continuity, a condition excluded by the hypothesis. 

And it immediately follows that the points (other than those infini 
tesimally near an essential singularity) in a region of continuity, at which a 
function assumes any the same value, are isolated points ; and that only a 
finite number of such points occur in any finite area. 

38. THEOREM II. The multiplicity m of any zero a of a function is 
finite provided the zero be an ordinary point of the function, which is not zero 
throughout its region of continuity; and the function can be expressed in the 



where <f> (z) is holomorphic in the vicinity of a, and a is not a zero of < (z). 

Let f(z) denote the function ; since a is a zero, we have f(a) = 0. 
Suppose that / (a), f" (a), ...... vanish: in the succession of the derivatives 

of f, one of finite order must be reached which does not have a zero value. 
Otherwise, if all vanish, then the function and all its derivatives vanish at a; 
the expansion of f(z) in powers of z a leads to zero as the value of f (z\ 
that is, the function is everywhere zero in the region of continuity, if all the 
derivatives vanish at a. 

Let, then, the wth derivative be the first in the natural succession which 
does not vanish at a, so that m is finite. Using Cauchy s expansion, we have 

(? n\tm) ( ~ _ n \(m+\) 

f(z) = (Z a / (a) + SZa_/F* (a) + . . . 

J m ! J (m + 1) ! J 

= (z-ay*$(z\ 

where < (z) is a function that does not vanish with a and, being the quotient 
of a converging series by a monomial factor, is holomorphic in the immediate 
vicinity of a. 

COROLLARY I. If infinity be a zero of a function of multiplicity m and 
at the same time be an ordinary point of the function, then the function can be 



expressed in the form z~ m $ f-J , 



where </>(-) is a function that is continuous and non-evanescent for infinitely 



large values of z. 

The result can be derived from the expansion in 30 in the same way as 
the foregoing theorem from Cauchy s expansion. 



62 ZEROS OF A [38. 

COROLLARY II. The number of zeros of a function, account being taken of 
their multiplicity, which occur within a finite area of the region of continuity 
of the function, is finite, when no point of the boundary of the area is infinitesi- 
mally near an essential singularity. 

By Corollary II. of 37, the number of distinct zeros in the limited area 
is finite, and, by the foregoing theorem, the multiplicity of each is finite ; 
hence, when account is taken of their respective multiplicities, the total 
number of zeros is still finite. 

The result is, of course, a known result for an algebraical polynomial ; but 
the functions in the enunciation are not restricted to be of the type of 
algebraical polynomials. 

Note. It is important to notice, both for the Theorem and for Corollary I, 
that the zero is an ordinary point of the function under consideration ; the 
implication therefore is that the zero is a definite zero and that in the 
immediate vicinity of the point the function can be represented in the form 

P(z a) or P [-] , the function P(a a) or P ( ) being .always a definite 

\<6 / \ / 

zero. 

Instances do occur for which this condition is not satisfied. The point 
may not be an ordinary point, and the zero value may be an indeterminate 
zero ; or zero may be only one of a set of distinct values though everywhere 
in the vicinity the function is regular. Thus the analysis of 13 shews that 

z=a is a point where the function sn - - has any number of zero values and 

Z CL 

any number of infinite values, and there is no indication that there are not 
also other values at the point. In such a case the preceding proposition does 
not apply ; there may be no limit to the order of multiplicity of the zero, and 
we certainly cannot infer that any finite integer m can be obtained such that 

(z - a)~ m <j> (z) 

is finite at the point. Such a point is ( 32) an essential singularity of the 
function. 

39. THEOREM III. A multiple zero of a function is a zero of its 
derivative ; and the multiplicity for the derivative is less or is greater by 
unity according as the zero is not or is at infinity. 

If a be a point in the finite part of the plane which is a zero of f(z) 
of multiplicity n, we have 

/(f)-(* T .a) + (X 

and therefore / (z) = (z - a) n ~ l [n$ (z} + (z-a) $ (z)}. 
The coefficient of (z a) n ~ l is holomorphic in the immediate vicinity of a and 
does not vanish for a ; hence a is a zero for / (z) of decreased multiplicity 



39.] UNIFORM FUNCTION 

If z = oo be a zero off(z) of multiplicity r, then 



where < (-) is holomorphic for very large values of z and does not vanish at 

\ z / 

infinity. Therefore 



The coefficient of ^~ r ~ 1 is holomorphic for very large values of z, and does 
not vanish at infinity ; hence z=<x> is a zero off (z) of increased multiplicity 
r + l. 

Corollary I. If a function be finite at infinity, then z = oo is a zero of the 
first derivative of multiplicity at least two. 

Corollary II. If a be a finite zero off(z) of multiplicity n, we have 
f(z) = n #(z) 
f(z) ir-** fW 

Now a is not a zero of <J> (z) ; and therefore ^4^r is finite, continuous, uniform 

9W 

and monogenic in the immediate vicinity of a. Hence, taking the integral 
of both members of the equation round a circle of centre a and of radius 
so small as to include no infinity and no zero, other than a, of / (z) _ and 
therefore no zero of $(z) we have, by 17 and Ex. 2, 25, 

~jT/ \ ^"^ ~ ^- 

/(*) 



40. THEOREM IV. A function must have an infinite value for some finite 
or infinite value of the variable. 

If M be a finite maximum value of the modulus for points in the plane, 
then ( 22) we have 



where r is the radius of an arbitrary circle of centre a, provided the whole of 
the circle is in the region of continuity of the function. But as the function 
is uniform, monogenic, finite and continuous everywhere, this radius can be 
increased indefinitely ; when this increase takes place, the limit of 

is zero and therefore /<> (a) vanishes. This is true for all the indices 1,2,... 
of the derivatives. 



64 INFINITIES OF A [40. 

Now the function can be represented at any point z in the vicinity of a 
by the series 



which degenerates, under the present hypothesis, to /(a), so that the function 
is everywhere constant. Hence, if a function has not an infinity somewhere 
in the plane, it must be a constant. 

The given function is not a constant; and therefore there is no finite 
limit to the maximum value of its modulus, that is, the function acquires 
an infinite value somewhere in the plane. 

COROLLARY I. A function must have a zero value for some finite or 
infinite value of the variable. 

For the reciprocal of a uniform monogenic analytic function is itself a 
uniform monogenic analytic function ; and the foregoing proposition shews 
that this reciprocal must have an infinite value for some value of the 
variable, which therefore is a zero of the function. 

COROLLARY II. A function must assume any assigned value at least once. 

COROLLARY III. Every function which is not a mere constant must have 
at least one singularity, either accidental or essential. For it must have 
an infinite value : if this be a determinate infinity, the point is an accidental 
singularity ( 32) ; if it be an infinity among a set of values at the point, the 
point is an essential singularity ( 32, 33). 

41. Among the infinities of a function, the simplest class is that con 
stituted by its accidental singularities, already defined ( 32) by the property 
that, in the immediate vicinity of such a point, the reciprocal of the function 
is regular, the point being an ordinary (zero) point for that reciprocal. 

THEOREM V. A function, which has a point cfor an accidental singularity, 
can be expressed in the foi*m 

(z - c}~ n (f> (z), 

where n is a finite positive integer and <f> (z) is a continuous function in the 
vicinity of c. 

Since c is an accidental singularity of the function f(z}, the function ^y-r 

/ ( z ) 

is regular in the vicinity of c and is zero there ( 32). Hence, by 38, there 
is a finite limit to the multiplicity of the zero, say n (which is a positive 
integer), and we have 



where ^ (z) is uniform, monogenic and continuous in the vicinity of c and is 
not zero there. The reciprocal of ^ (z), say <f> (z), is also uniform, monogenic 



41.] UNIFORM FUNCTION 65 

and continuous in the vicinity of c, which is an ordinary point for (f> (z) ; 
hence we have 

f(z} = ( Z -c)-^(z\ 

which proves the theorem. 

The finite positive integer n measures the multiplicity of the accidental 
singularity at c, which is sometimes said to be of multiplicity n or of 
order n. 

Another analytical expression for f(z) can be derived from that which 
has just been obtained. Since c is an ordinary point for <f> (z) and not a zero, 
this function can be expanded in a series of ascending, positive, integral 
powers of z c, converging in the vicinity of c, in the form 

(*) = P(*-c) 

= u Q + u l (z-c} + ... + u n ^(z-c) n - l +u n (z-c) n +... 

= u + u,(z - c) + ... + u n _^(z - c) 71 - 1 + (z- c) n Q(z-c), 
where Q(z c), a series of positive, integral, powers of z c converging in the 
vicinity of c, is a monogenic analytic function of z. Hence we have 

^ = ^ + ( 7^+ - +,~; + ( - )> 

the indicated expression for f(z), valid in the immediate vicinity of c, where 
Q (z c) is uniform, finite, continuous and monogenic. 

COROLLARY. A function, which has z= oo for an accidental singularity of 
multiplicity n, can be expressed in the form 



_ 

where </>(-) is a continuous function for very large values of \z , and is not 

\zj 

zero when z = oo . It can also be expressed in the form 

1 + ... + a n ^ z + Q (-} , 
\zj 



where Q ( - j is uniform, finite, continuous and monogenic for very large values 

f\\. 

The derivation of the form of the function in the vicinity of an accidental 
singularity has been made to depend upon the form of the reciprocal of the 
function. Whatever be the (finite) order of that point as a zero of the 
reciprocal, it is assumed that other zeros of the reciprocal are not at merely 
infinitesimal distances from the point, that is, that other infinities of the 
function are not at merely infinitesimal distances from the point. 

Hence the accidental singularities of a function are isolated points ; and 
there is only a finite number of them in any limited portion of the plane. 
F. 5 



66 INFINITIES OF A [42. 

42. We can deduce a criterion which determines whether a given singu 
larity of a function /(f) is accidental or essential. 

When the point is in the finite part of the plane, say at c, and a finite 
positive integer n can be found such that 

is not infinite at c, then c is an accidental singularity. 

When the point is at infinity and a finite positive integer n can be found 
such that 

is not infinite when z = oc , then z = oo is an accidental singularity. 

If one of these conditions be not satisfied, the singularity at the point is 
essential. But it must not be assumed that the failure of the limitation to 
finiteness in the multiplicity of the accidental singularity is the only source 
or the complete cause of essential singularity. 

Since the association of a single factor with the function is effective in 
preventing an infinite value at the point when one of the conditions is 
satisfied, it is justifiable to regard the discontinuity of the function at 
the point as not essential and to call the singularity either non-essential 
or accidental ( 82). 

43. THEOREM VI. The poles of a function, that lie in the finite part 
of the plane, are all the poles (of increased multiplicity) of the derivatives of 
the function that lie in the finite part of the plane. 

Let c be a pole of the function f(z) of multiplicity p : then, for any point 
z in the vicinity of c, 

where </> (z) is holomorphic in the vicinity of c, and does not vanish for z = c. 
Then we have 

f ( 2 ) = ( z ~ c )~ p $ ( z ) ~ P ( 2 ~ c ) p 1 $ W 
= (z-c)-P-*{(z-c)<j> (z)-p<}>(z)} 

where % (z) is holomorphic in the vicinity of c, and does not vanish for z = c. 

Hence c is a pole of/ (z) of multiplicity ^9 + 1. Similarly it can be shewn 
to be a pole of / (r) (z) of multiplicity p + r. 

This proves that all the poles of f(z) in the finite part of the plane are 
poles of its derivatives. It remains to prove that a derivative cannot have 
a pole which the original function does not also possess. 

Let a be a pole off (z) of multiplicity m : then, in the vicinity of a,f (z) 
can be expressed in the form 



43.] UNIFORM FUNCTION 7 

where ^ ( z ) is holomorphic in the vicinity of a and does not vanish for z = a 
Thus 



and therefore f (*) = - . + j_ , 

y v JlV^ <*-)*"* 

so that, integrating, we have 

f(z}= *() _*>) 

m - a)- 1 (m - 1) - a)- 2 
that is, a is a pole of/0). 

An apparent exception occurs in the case when m is unity: for then 
we have 



the integral of which leads to 

f(z} = ^ (a) log (z - a) + . . . , 

so that/0) is no longer uniform, contrary to hypothesis. Hence a derivative 
cannot have a simple pole in the finite part of the plane ; and so the exception 
is excluded. 

The theorem is thus proved. 

COROLLARY I. The r th derivative of a function cannot have a pole in the 
finite part of the plane of multiplicity less than r + 1. 

COROLLARY II. If c be a pole of f (z) of any order of multiplicity ^ and 
if f (r] (z) be expressed in the form 

, _ Oi__ 

/ _. _\., _!_* _ 1 I * 



(Z - CY +T (Z- 

there are no terms in this expression with the indices - 1, - 2, ...... , - r. 

COROLLARY III. If c be a pole of/ (z) of multiplicity p, we have 



= 
f(z) z-c~* 4>(z) 

where $ (z) is a holomorphic function that does not vanish for z = c, so that 

< 0) 

-T-/JN is a holomorphic function in the vicinity of c. Taking the integral of 

f (z) 

-j-j~\ roun d a circle, with c for centre, with radius so small as to exclude all 

other poles or zeros of the function f (z), we have 



52 



(}8 INFINITIES OF A [43. 

COROLLARY IV. If a simple closed curve include a number N of zeros of 
a uniform function f (z) and a number P of its poles, in both of which 
numbers account is taken of possible multiplicity, and if the curve contain 
no essential singularity of the function, then 



the integral being taken round the curve. 

f (z) 
The only infinities of the function ^i within the curve are the zeros 

j( z ) 

and the poles of / (z). Round each of these draw a circle of radius so small 
as to include it but no other infinity ; then, by Cor. II. 18, the integral 
round the closed curve is the sum of the values when taken round these 
circles. By the Corollary II. 39 and by the preceding Corollary III., the 
sum of these values is 

= 2w %> 

= N-P. 

It is easy to infer the known theorem that the number of roots of an 
algebraical polynomial of order n is n, as well as the further result that 
2^ (N - P) is the variation of the argument of / (z) as z describes the 
closed curve in a positive sense. 

Ex. Prove that, if F(z) be holomorphic over an area, of simple contour, which con 
tains roots !, 2 ,... of multiplicity m m 2 ,... and poles c x , c 2) ... of multiplicity p^ p 2J ... 
respectively of a function f(z) which has no other singularities within the contour, then 



the integral being taken round the contour. 

In particular, if the contour contains a single simple root a and no singularity, then that 
root is given by 



the integral being taken as before. (Laurent.) 

44. THEOREM VII. If infinity be a pole of f (z), it is also a pole of 
f (z) only when it is a multiple pole of f (z). 

Let the multiplicity of the pole for f (z) be ?i; then for very large values 
of z we have 

/(*)-**), 

where <j> is holomorphic for very large values of z and does not vanish at 
infinity ; hence 

A)**" *-* 






44.] UNIFORM FUNCTION 69 

The coefficient of z n ~* is holomorphic for very large values of z and does not 
vanish at infinity ; hence infinity is a pole of/ (z} of multiplicity n 1. 

If n be unity, so that infinity is a simple pole of / (z), then it is not a 
pole of/ (2); the derivative is then finite at infinity. 

45. THEOREM VIII. A function, which has no singularity in a finite 
part of the plane, and has z = oo for a pole, is an algebraical polynomial. 

Let n, necessarily a finite integer, be the order of multiplicity of the pole 
at infinity : then the function / (z) can be expressed in the form 



1 + ...... +a n ^z + Q - , 

\zJ 

where Q (- J is a holomorphic function for very large values of z, and is finite 
(or zero) when z is infinite. 

Now the first n terms of the series constitute a function which has no 
singularities in the finite part of the plane : and / (z) has no singularities 

in that part of the plane. Hence Q ( - J has no singularities in the finite part 

of the plane : it is finite for infinite values of z. It thus can never have an 
infinite value: and it is therefore merely a constant, say a n . Then 

/ (z) = a,z n + a^- 1 + ...... + a n ^z + a n , 



a polynomial of degree equal to the multiplicity of the pole at infinity, 
supposed to be the only pole of the function. 

46. The above result may be obtained in the following manner. 

Since z = GO is a pole of multiplicity n, the limit of z~ n f (z} is not infinite 
when z = oo . 

Now in any finite part of the plane the function is everywhere finite, so 
that we can use the expansion 



where = * ""> dt 



+l t-z 

the integral being taken round a circle of any radius r enclosing the point z 
and having its centre at the origin. As the subject of integration is finite 
everywhere along the circumference, we have, by Darboux s expression in 
(IV.) S 14, 



T i T _ z 

where r is some point on the circumference and X is a quantity of modulus 
not greater than unity. 



70 TRANSCENDENTAL AND [46. 

Let T = re ia - ; then 



X . fM 

" 71-4-1 flii / \ / 

?* r n 



r 



f( T \ 
By definition, the limit of n as T (and therefore r) becomes infinitely 

( -\ 1 
1 -- e~ ai } is unity. 
r J 

Since \ is not greater than unity, the limit of \jr in the same case is zero ; 
hence with indefinite increase of r, the limit of R is zero and so 



shewing as before that/(^) is an algebraical polynomial. 

47. As the quantity n is necessarily a positive integer*, there are two 
distinct classes of functions discriminated by the magnitude of n. 

The first (and the simpler) is that for which n has a finite value. The 
polynomial then contains only a finite number of terms, each with a positive 
integral index ; and the function is then a rational, integral, algebraical 
polynomial of degree n. 

The second (and the more extensive, as significant functions) is that 
for which n has an infinite value. The point z = oo is not a pole, for then 
the function does not satisfy the test of 42 : it is an essential singularity 
of the function, which is expansible in an infinite converging series 
of positive integral powers. To functions of this class the general term 
transcendental is applied. 

The number of zeros of a function of the former class is known : it is 
equal to the degree of the function. It has been proved that the zeros of a 
transcendental function are isolated points, occurring necessarily in finite 
number in any finite part of the region of continuity of the function, no 
point on the boundary of the part being infinitesimally near an essential 
singularity ; but no test has been assigned for the determination of the total 
number of zeros of a function in an infinite part of the region of con 
tinuity. 

Again, when the zeros of a polynomial are given, a product-expression can 
at once be obtained that will represent its analytical value. Also we know 
that, if a be a zero of any uniform analytic function of multiplicity n, the 
function can be represented in the vicinity of a by the expression 

(x-a} n <t>(z\ 

where < (z) is holomorphic in the vicinity of a. The other zeros of the 
function are zeros of <f> (z) ; this process of modification in the expression 

* It is unnecessary to consider the zero value of n, for the function is then a polynomial of 
order zero, that is, it is a constant. 



47.] ALGEBRAICAL UNIFORM FUNCTIONS 71 

can be continued for successive zeros so long as the number of zeros taken 
account of is limited. But when the number of zeros is unlimited, then the 
inferred product-expression for the original function is not necessarily a 
converging product; and thus the question of the formal factorisation of a 
transcendental function arises. 

48. THEOREM IX. A function, all the singularities of which are accid 
ental, is a rational, algebraical, meromorphic function. 

Since all the singularities are accidental, each must be of finite 
multiplicity ; and therefore infinity, if an accidental singularity, is of finite 
multiplicity. All the other poles are in the finite part of the plane ; they 
are isolated points and therefore only finite in number, so that the total 
number of distinct poles is finite and each is of finite order. Let them be 
!, a 2 , ...... , a^ of orders m 1} m 2 , ...... , m^ respectively : let m be the order of 

the pole at infinity: and let the poles be arranged in the sequence of 
decreasing moduli such that [aj > a F _! > ...... >|&i|- 

Then, since infinity is a pole of order m, we have 

/ 0) = a m z m + a^z- 1 + ...... + a^z + / <, 

where / (z) is not infinite for infinite values of z. Now the polynomial 

m 

Sttj^ is not infinite for any finite value of z ; hence f (z) is infinite for all 

i = l 

the finite infinities of f (z) and in the same way, that is, the function f (z) 
has !, ...... , a^ for its poles and it has no other singularities. 

Again, since M is a finite pole of multiplicity W M , we have 



where fi(z) is not infinite for z = a ll and, as f (z) is not infinite for z=<x> , 
evidently f^ (z) is not infinite for z = oo . Hence the singularities of f^ (z) are 
merely the poles a 1} ...... , a F _i ; and these are all its singularities. 

Proceeding in this manner for the singularities in succession, we ultimately 
reach a function f^ (z) which has only one pole a^ and no other singularity, 
so that 

k k 



where g (z) is not infinite for z = a^ But the function f^(z) is infinite only 
for 2 = 0,!, and therefore g (2) has no infinity. Hence g (z} is only a constant, 
say k : thus 

9 (*} = ^o- 

Combining all these results we have a, finite number of series to add together: 
and the result is that 



72 UNIFORM [48. 

where g 1 (z) is the series k + a-^z + + a m z m , and \ I is the sum of the 

finite number of fractions. Evidently g s (z) is the product 

{z Oi) m> (z a 2 ) ma (z a M ) m fx ; 

and g (z) is at most of degree 



If F (z} denote g 1 (z} g 3 (z) + g^ (z), the form of / (z) is 



</.(*) 

that is, f (z) is a rational, algebraical, meromorphic function. 

It is evident that, when the function is thus expressed as an algebraical 
fraction, the degree of F (z) is the sum of the multiplicities of all the poles 
when infinity is a pole. 

COROLLARY I. A function, all the singularities of which are accidental, 
has as many zeros as it has accidental singularities in the plane. 

If z = oo be a pole, then it follows that, because f(z) can be expressed 
in the form 



it has as many zeros as F(z), unless one such should be also a zero of g^(z). 
But the zeros of g 3 (z) are known, and no one of them is a zero of F(z), on 
account of the form of f(z} when it is expressed in partial fractions. Hence 
the number of zeros off(z) is equal to the degree of F(z}, that is, it is equal 
to the number of poles off(z}. 

If 2=00 be not a pole, two cases are possible; (i) the function f (z) may be 
finite for z = oo , or (ii) it may be zero for z = oo . In the former case, the 
number of zeros is, as before, equal to the degree of F (z), that is, it is equal 
to the number of infinities. 

In the latter case, if the degree of the numerator F (z) be K less than 
that of the denominator g s (z), then z = oo is a zero of multiplicity K ; and it 
follows that the number of zeros is equal to the degree of the numerator 
together with K, so that their number is the same as the number of accidental 
singularities. 

COROLLARY II. At the beginning of the proof of the theorem of the 
present section, it is proved that a function, all the singularities of which are 
accidental, has only a finite number of such singularities. 

Hence, by the preceding Corollary, such a function can have only a finite 
number of zeros. 

If, therefore, the number of zeros of a function be infinite, the function 
must have at least one essential singularity. 



48.] ALGEBRAICAL FUNCTIONS 73 

COROLLARY III. When a uniform analytic function has no essential 
singularity, if the (finite) number of its poles, say c lv .., c m , be m, no one of 
them being at z = oo , and if the number of its zeros, say a ly ..., a m , be also m, 
no one of them being at z = oo , then the function is 



n 

* a 



r=l \Z - C T 

except possibly as to a constant factor. 

When z = oo is a zero of order n, so that the function has m n zeros, say 
i, a 2 ,..., in the finite part of the plane, the form of the function is 



m-n 

II (z a r ) 

r=l 



r=l 



and, when z = <x> is a pole of order p, so that the function has m - p poles, 
say c l} c. 2> ..., in the finite part of the plane, the form of the function is 



II (Z - Or) 

r=l _ 

m-p ~ 



COROLLARY IV. All the singularities of rational algebraical meromorphic 
functions are accidental. 



CHAPTER V. 

TRANSCENDENTAL INTEGRAL FUNCTIONS. 

49. WE now proceed to consider the properties of uniform functions 
which have essential singularities. 

The simplest instance of the occurrence of such a function has already 
been referred to in 42 ; the function has no singularity except at z = oo , 
and that value is an essential singularity solely through the failure of the 
limitation to finiteness that would render the singularity accidental. The 
function is then an integral function of transcendental character ; and it is 
analytically represented ( 26) by G (z) an infinite series in positive powers of 
z, which converges everywhere in the finite part of the plane and acquires 
an infinite value at infinity alone. 

The preceding investigations shew that uniform functions, all the singu 
larities of which are accidental, are rational algebraical functions their 
character being completely determined by their uniformity and the accidental 
nature of their singularities, and that among such functions having the same 
accidental singularities the discrimination is made, save as to a constant 
factor, by means of their zeros. 

Hence the zeros and the accidental singularities of a rational algebraical 
function determine, save as to a constant factor, an expression of the function 
which is valid for the whole plane. A question therefore arises how far 
the zeros and the singularities of a transcendental function determine the 
analytical expression of the function for the whole plane. 

50. We shall consider first how far the discrimination of transcendental 
integral functions, which have no infinite value except for z = oc , is effected 
by means of their zeros*. 

* The following investigations are based upon the famous memoir by Weierstrass, " Zur 
Theorie der eindeutigen analytischen Functionen," published in 187G : it is included, pp. 1 52, 
in the Abhandlungen aus der Functioiienlehre (Berlin, 1886). 

In connection with the product-expression of a transcendental function, Cayley, " Memoire sur 
les fonctions doublement periodiques," Liouville, t. x, (1845), pp. 385 420, or Collected Works, 
vol. i, pp. 156 182, should be consulted. 



50.] 



CONVERGING INFINITE PRODUCTS 



75 



Let the zeros a ly a 2 , a 3 ,... be arranged in order of increasing moduli; a 
finite number of terms in the series may have the same value so as to allow 
for the existence of a multiple zero at any point. After the results stated 
47, it will be assumed that the number of zeros is infinite ; that, 



n 



subject to limited repetition, they are isolated points ; and, in the present 
chapter, that, as n increases indefinitely, the limit of \a n \ is infinity. And it 
will be assumed that a t \ > 0, so that the origin is temporarily excluded from 
the series of zeros. 

Let z be any point in the finite part of the plane. Then only a limited 
number of the zeros can lie within and on a circle centre the origin and 
radius equal to \z\ ; let these be a ]5 a 2 ,..., a fc _ 1} and let a r denote any one of 
the other zeros. We proceed to form the infinite product of quantities u r , 
where u r denotes 



and g r is a rational integral function of z which, being subject to choice, will 
be chosen so as to make the infinite product converge everywhere in the 
plane. We have 

00 \ 



w=l 

a series which converges because \z < \a r \. Now let 

ffr = 

then 



> 1 / ^ \n 
i v - 1 - / * \ 

logi< r = - 2 -fJ , 

j = S 4 \**rr 



and therefore 



Hence 



- " 



if the expression on the right-hand side be finite, that is, if the series 

oo ce I / _ \ n 

2 S -(-) 

r=ftw=^ \flrf 

converge unconditionally. Denoting the modulus of this series by M, we 
have 

z 
a,. 



00 00 1 

M < 2 2 - 

r-k n=s M 



SO that 



sM< S 2 

r=k n=s 




7G 



WEIERSTRASS S CONVERGING 



[50. 



whence since 1 - is the smallest of the denominators in terms of the last 

* 
sum, we have 



sM\l- 


z 


[ < 


00 




Z 


8 


1 


& 


j r=k 


a r 








I l 







*-l 





If, as is not infrequently the case, there be any finite integer s for which (and 
therefore for all greater indices) the series 

2 1 



Is 



00 

and therefore the series 2 \a r \- s , converges, we choose s to be that least 

r=k 

integer. The value of M then is finite for all finite values of z ; the series 



oo co T / ~\n 

2 2 - - 



n 

r=k 



converges unconditionally and therefore 
is a converging product when 

Let the finite product 

A-l (/ f 

n |(i-- 

m=l l\ a m 

be associated as a factor with the foregoing infinite converging product. Then 
the expression 

oo ( f 2 \ 2 

T-=I (\ a r / 
is an infinite product, converging uniformly and unconditionally for all finite 

00 

values of z, provided the finite integer s be such as to make the series 2 






converge uniformly and unconditionally. 

Since the product converges uniformly and unconditionally, no product 
constructed from its factors u r , say from all but one of them, can be infinite. 
Now the factor 

"5?i/-Y 

\ ?L\ e n=\n\a m ) 



vanishes for z = a m ; hence f(z) vanishes for z = a m . Thus the function, 
evidently uniform after what has been proved, has the assigned points 
Oj, a 2) ... and no others for its zeros. 



50.] 



INFINITE PRODUCT 



77 



Further, z = oo is an essential singularity of the function ; for it is an 
essential singularity of each of the factors on account of the exponential 
element in the factor. 

51. But it may happen that no finite integer s can be found which will 
make the series 

00 

r=l 

converge*. We then proceed as follows. 

Instead of having the same index s throughout the series, we associate 
with every zero a r an integer m r chosen so as to make the series 



n=l @"n \Q"n 

a converging series. To obtain these integers, we take any series of decreasing 
real positive quantities e, e 1} e 2 ,..., such that (i) e is less than unity and 
(ii) they form an unconditionally converging series ; and we choose integers 
fti r such that 



These integers make the foregoing series of moduli converge. For, 
neglecting the limited number of terms for which \z\^ a\, and taking e 
such that 

z 



we have for all succeeding terms 



and therefore 



a r 



Hence, except for the first k 1 terms, the sum of which is finite, we have 



n=k 



which is finite because the series 



... converges. Hence the series 



n=l 



s a converging series. 



* For instance, there is no finite integer s that can make the infinite series 
(log 2)- + (log 3)- + (log 4)- + . . . 

converge. This series is given in illustration by Hermite, Cours a la faculte des Sciences (4 mc ed. 
1891), p. 86. 



78 



WEIERSTRASS S CONVERGING 



[51. 



Just as in the preceding case a special expression was formed to serve as 
a typical factor in the infinite product, we now form a similar expression 
for the same purpose. Evidently 



1 - a; = e i<* a-*) = e 
if \x\ < 1. Forming a function E (x, m) denned by the equation 



m x r 

S - 



E (x, m)=(l-x)e r=1 r , 



we have E (x, m) = 

In the preceding case it was possible to choose the integer m so that it 
should be the same for all the factors of the infinite product, which was 



ultimately proved to converge. Now, we take x = and associate m n as 
the corresponding value of m. Hence, if 

/(*) = 



where 



< \z < |ttjfc|, we have 



n=k 



- s s 



The infinite product represented by f(z) will converge if the double series in 
the exponential be a converging series. 

Denoting the double series by S, we have 



\S\<* 2 



2^* 
2 

n=kr=l 



r+m n 



< 2 

nk 



1+TO M 



1 4 

\a n 



on effecting the summation for r. Let A be the value of 1 



all the remaining values of n we have 

1 



z ! 



- ; then for 



>>A, 



and so 



n=/fc 



This series converges; hence for finite values of z\ the value of \S\ is 
finite, so that S is a converging series. Hence it follows that f(z) is an 



51.] INFINITE PRODUCT 79 

unconditionally converging product. We now associate with f(z) as factors 
the k I functions 



for i= 1, 2,..., k1; their number being finite, their product is finite and 
therefore the modified infinite product still converges. We thus have 



an unconditionally converging product. 

Since the product G (z) converges unconditionally, no product constructed 
from its factors E, say from all but one of them, can be infinite. The factor 



vanishes for the value z = a n and only for this value ; hence G (z) vanishes for 
z = a n . It therefore appears that G(z) has the assigned points a 1} a.,, a 3 , ... 
and no others for its zeros ; and from the existence of the exponential in each 
of the factors it follows that z = oo is an essential singularity of the factor and 
therefore it is an essential singularity of the function. 

Denoting the series in the exponential by g n (z\ so that 

m n 1 / ~ \ r 

*<*>-?() 

71 / z \ i-. Z\ 

we have A , m n = 1 e^ ; 

\a n / V aJ 

and therefore the function obtained is 

; G (z)= H \(l ] e g (zl 

n = l (\ Q"n,l 

The series g n usually contains only a limited number of terms ; when the 
number of terms increases without limit, it is only with indefinite increase 
of | a n | and the series is then a converging series. 

It should be noted that the factors of the infinite product G (z) are the 
expressions E no one of which, for the purposes of the product, is resoluble 
into factors that can be distributed and recombined with similarly obtained 
factors from other expressions E; there is no guarantee that the product 
of the factors, if so resolved, would converge uniformly and unconditionally, 
and it is to secure such convergence that the expressions E have been 
constructed. 

It was assumed, merely for temporary convenience, that the origin was not 
a zero of the required function ; there obviously could not be a factor of 
exactly the same form as the factors E if a were the origin. 



80 TRANSCENDENTAL INTEGRAL FUNCTION [51. 

If, however, the origin were a zero of order X, we should have merely 
to associate a factor Z K with the function already constructed. 

We thus obtain Weierstrass s theorem : 

It is possible to construct a transcendental integral function such that it 
shall have infinity as its only essential singularity and have the origin (of 
multiplicity X), a^, a z , a 3 , ... as zeros ; and such a function is 

00 ( / z\ 

Z K n ui U^ 



n=i 

where g n (z) is a rational, integral, algebraical function of z, the form of which 
is dependent upon the law of succession of the zeros. 

52. But, unlike uniform functions with only accidental singularities, the 
function is not unique : there are an unlimited number of transcendental 
integral functions with the same series of zeros and infinity as the sole essential 
singularity, a theorem also due to Weierstrass. 

For, if G! (z) and G (z) be two transcendental, integral functions with the 
same series of zeros in the same multiplicity, and z = oo as their only essential 
singularity, then 



G(z} 
is a function with no zeros and no infinities in the finite part of the plane. 

Denoting it by r 2 , then 

1 ^ 

<7 2 dz 

is a function which, in the finite part of the plane, has no infinities; and 
therefore it can be expanded in the form 



a series converging everywhere in the finite part of the plane. Choosing a 
constant C so that 6r 2 (0) = e* 7 ", we have on integration 



where g(z) = C 

and g (z) is finite everywhere in the finite part of the plane. Hence it follows 
that, ifg(z) denote any integral function of z which is finite everywhere in the 
finite part of the plane, and if G (z) be some transcendental integral function 
with a given series of zeros and z= oo as its sole essential singularity, all 
transcendental integral functions with that series of zeros and z= <x> as the 
sole essential singularity are included in the form 

(*)*. 

COROLLARY I. A function which has no zeros in the finite part of the 
plane, no accidental singularities and z=<x> for its sole essential singularity 
is necessarily of the form 



52.] AS AN INFINITE PRODUCT 81 

where g (z) is an integral function of z finite everywhere in the finite part 
of the plane. 

COROLLARY II. Every transcendental function, which has the same zeros 
in the same multiplicity as an algebraical polynomial A (z) the number, 
therefore, being necessarily finite , ivhich has no accidental singularities and 
has z = oo for its sole essential singularity, can be expressed in the form 

A (z) 



COROLLARY III. Every function, which has an assigned series of zeros 
and an assigned series of poles and has z = oo for its sole essential singu 
larity, is of the form 



where the zeros of G (z) are the assigned zeros and the zeros of G p (z) are the 
assigned poles. 

For if Op (z) be any transcendental integral function, constructed as in 
the proposition, which has as its zeros the poles of the required function in 
the assigned multiplicity, the most general form of that function is 

p (*)e*, 

where h (z) is integral. Hence, if the most general form of function which 
has those zeros for its poles be denoted by f(z), we have 

f(z)G p (z)e^ 

as a function with no poles, with infinity as its sole essential singularity, and 
with the assigned series of zeros. But if G (z) be any transcendental integral 
function with the assigned zeros as its zeros, the most general form of function 
with those zeros is 



and so f(z) G p (z) e h = G (z) e & , 

whence / (z) = ?$1 e ffW, 

Lr p (z) 

in which g (z) denotes g (z) h (z). 

If the number of zeros be finite, we evidently may take G (z) as the 
algebraical polynomial with those zeros as its only zeros. 

If the number of poles be finite, we evidently may take G p (z) as the 
algebraical polynomial with those poles as its only zeros. 

And, lastly, if a function have a finite number of zeros, a finite number 
of accidental singularities and 2=00 as its sole essential singularity, it can 
be expressed in the form 



F. 



82 PRIMARY [52. 

where P and Q are rational integral polynomials. This is valid even though 
the number of assigned zeros be not the same as the number of assigned 
poles ; the sole effect of the inequality of these numbers is to complicate the 
character of the essential singularity at infinity. 

53. It follows from what has been proved that any uniform function, 
having z = <x> for its sole essential singularity and any number of assigned 
zeros, can be expressed as a product of expressions of the form 



a 



Such a quantity is called* a primary factor of the function. 

It has also been proved that : 

(i) If there be no zero a n , the primary factor has the form 

(ii) The exponential index g n (z) may be zero for individual primary 
factors, though the number of such factors must, at the utmost, 
be finite f. 

(iii) The factor takes the form z when the origin is a zero. 
Hence we have the theorem, due to Weierstrass : 

Every uniform integral function of z can be expressed as a product of 
primary factors, each of the form 

(kz + I) e3W, 

where g(z) is an appropriate integral function of z vanishing with z and where 
k, I are constants. In particular factors, g (z) may vanish ; and either k or I, 
but not both k and I, may vanish with or without a non-vanishing exponential 
index g(z). 

54. It thus appears that an essential distinction between transcendental 
integral functions is constituted by the aggregate of their zeros : and we may 
conveniently consider that all such functions are substantially the same when 
they have the same zeros. 

There are a few very simple sets of functions, thus discriminated by their 
zeros: of each set only one member will be given, and the factor e^ (z} , which 
makes the variation among the members of the same set, will be neglected 
for the present. Moreover, it will be assumed that the zeros are isolated 
points. 

I. There may be a finite number of zeros ; the simplest function is then 
an algebraical polynomial. 

* Weierstrass s term is Prim/unction, I.e., p. 15. 

t Unless the class ( 59) be zero, when the index is zero for all the factors. 



54.] FACTORS 83 

II. There may be a singly-infinite system of zeros. Various functions 
will be obtained, according to the law of distribution of the zeros. 

Thus let them be distributed according to a law of simple arithmetic 
progression along a given line. If a be a zero, co a quantity such that co \ 
is the distance between two zeros and arg. co is the inclination of the line, 
we have 

a + mco, 

for integer values of m from - oo to + oo , as the expression of the series of 
the zeros. Without loss of generality we may take a at the origin this 
is merely a change of origin of coordinates and the origin is then a 
simple zero : the zeros are given by mco, for integer values of m from 
oo to + oo . 

Now 2 - = - 2 is a diverging series ; but an integer s the lowest 

value is s = 2 can be found for which the series S I - ] converges uni- 

\mcoj 
formly and unconditionally. Taking s = 2, we have 

, . - 1 1 / z \ n z 
ff m (z) = 2 - = , 
=i n vW m 
so that the primary factor of the present function is 



Z \ 

--- ) 
mco/ 



m<a 

e 



and therefore, by 52, the product 

/-,SJ(i- *-) 

-oo (\ mcoj 
converges uniformly and unconditionally for all finite values of z. 

The term corresponding to m = is to be omitted from the product ; and 
it is unnecessary to assume that the numerical value of the positive infinity 
for m is the same as that of the negative infinity for m. If, however, the 
latter assumption be adopted, the expression can be changed into the ordinary 
product-expression for a sine, by combining the primary factors due to values 
of m that arc equal and opposite : in fact, then 



co . TTZ 
= - - sin . 

7T CO 



This example is sufficient to shew the importance of the exponential term in the 
primary factor. If the product be formed exactly as for an algebraical polynomial, then 
the function is 



z n 

in the limit when both p and q are infinite. But this is known* to be 



- ) - sin . 

77 0) 



* Hobson s Trigonometry, 287. 

62 



84 PRIMARY [54. 

Another illustration is afforded by Gauss s II-function, which is the limit when k is 
infinite of 

1.2.3 ...... k 

(+!) (0+2) ...... (z+k) 

This is transformed by Gauss* into the reciprocal of the expression 



that is, of (1 +*) jj {(l +^) e " 2l g 

the primary factors of which have the same characteristic form as in the preceding 
investigation, though not the same literal form. 

It is chiefly for convenience that the index of the exponential part of the primary 

t-l 1/2 \n 

factor is taken, in 50, in the form 2 - ( ) . With equal effectiveness it may be 

n=l % \~^T / 
-l 1 

taken in the form 2 - b r n z n . provided the series 




r=k =i n 
converge uniformly and unconditionally. 

Ex. 1. Prove that each of the products 






form=+l, 3, +5, ...... to infinity, and 



the term for n = Q being excluded from the latter product, converges uniformly and uncon 
ditionally and that each of them is equal to cos z. (Hermite and Weyr.) 

Ex. 2. Prove that, if the zeros of a transcendental integral function be given by the 
series 

0) +&>, 4w, +9cB, ...... to infinity, 

the simplest of the set of functions thereby determined can be expressed in the form 

( fz\*\ , (. fz\*\ 
sm X?r I - }- sin -UTT - ) }- . 
I W ) ( W J 

Ex. 3. Construct the set of transcendental integral functions which have in common 
the scries of zeros determined by the law m 2 a> l + 2m<a 2 + a> 3 for all integral values of m 
between - oo and + oo ; and express the simplest of the set in terms of circular functions, j 

55. The law of distribution of the zeros, next in importance and sub 
stantially next in point of simplicity, is that in which the zeros form a doubly- 
infinite double arithmetic progression, the points being the oo 2 intersections 
of one infinite system of equidistant parallel straight lines with another 
infinite system of equidistant parallel straight lines. 

The origin may, without loss of generality, be taken as one of the zeros. 
If a) be the coordinate of the nearest zero along the line of one system 
passing through the origin, and &> be the coordinate of the nearest zero along 

* Ges. Wcrke, t. Hi, p. 145; the example is quoted in this connection by Weierstrass, I.e., ! 
p. 15. 



55.] FACTIOUS 85 

the line of the other system passing through the origin, then the complete 
series of zeros is given by 

fl = mw + mm, 

for all integral values of m and all integral values of ni between <x> and 
+ oo . The system of points may be regarded as doubly -periodic, having &> 
arid &> for periods. 

It must be assumed that the two systems of lines intersect. Other 
wise, w and to would have the same argument and their ratio would be a real 
quantity, say a ; and then 

ft 

= m + m a. 

CO 

Whether a be commensurable or incommensurable, the number of pairs 
of integers, for which m + in a. is zero or may be made less than any small 
quantity 8, is infinite ; and in either case we should have the origin a zero 
for each such pair, that is, altogether the origin would be a zero of infinite 
multiplicity. This property of a function is to be considered as excluded, 
for it would make the origin an essential singularity instead of, as required, 
an ordinary point of the transcendental integral function. Hence the ratio of 
the quantities w and w is not real. 

56. For the construction of the primary factor, it is necessary to render 
the series 



converging, by appropriate choice of integers s m>m . It is found to be 
possible to choose an integer s to be the same for every term of the series, 
corresponding to the simpler case of the general investigation, given in 50. 
As a matter of fact, the series 

diverges for s = I (we have not made any assumption that the positive and 
the negative infinities for m are numerically equal, nor similarly as to m ) ; 
the series converges for s = 2, but its value depends upon the relative values 
of the infinities for m and m ; and s = 3 is the lowest integral value for which, 
as for all greater values, the series converges uniformly and unconditionally. 

There are various ways of proving the uniform and unconditional conver 
gence of the series 2ft~ M when /* > 2 : the following proof is based upon a 
general method due to Eisenstein*. 



I=QO n=oo 



First, the series S 2 (m 2 + n*)~* converges uniformly and uricondi- 

m= > n= -oo 

tionally, if /j,> 1. Let the series be arranged in partial series : for this purpose, 



Crelle, t. xxxv, (1847), p. 161 ; a geometrical exposition is given by Halphen, Traite des 
fonctions elliptiques, t. i, pp. 358 362. 



86 WEIERSTRASS S FUNCTION AS [56. 

we choose integers k and I, and include in each such partial series all 
the terms which satisfy the inequalities 

m ^ 2* +1 , 



so that the number of values of m is 2* and the number of values of n is 2*. 
Then, if k + I = %K, we have 



so that each term in the partial series ^ ^- . The number of terms in the 

^" J* 

partial series is 2 fc . 2*, that is, 2 2K : so that the sum of the terms in the 
partial series is 



Take the upper limit of k and I to be p, ultimately to be made infinite. 
Then the sum of all the partial series is 



which, when p = oo , is a finite quantity if p > 1. 
Next, let (a = a. + /3i, = 7 + Si, so that 

ft = mw + nay = ma + ny + i (m{3 + n8) ; 
hence, if 6 = ma. + nj, (j> = m(3 + n$, 

we have | ft 2 = fr + </> 2 . 

Now take integers r and s such that 

r<0<r + \, s<(jxs + ~L. 

The number of terms ft satisfying these conditions is definitely finite and is 
independent of m and n. For since 

m(S 



n a - 

and a8 (3y does not vanish because o> /a> is not purely real, the number of 
values of in is the integral part of 

(r + 1)8 sy 

a.8 fiy 
less the integral part of 

r8 (s + 1 ) 7 
a.8 fly 

that is, it is the integral part of (7 + 8)/(8 #7). Similarly, the number of 
values of n is the integral part of (a + /3)/(aS - j3j). Let the product of the 



56.] A DOUBLY-INFINITE PRODUCT 87 

last two integers be q ; then the number of terms fl satisfying the in 
equalities is q. 

Then 22 1 ft \~* = 22 (&> + p)~* 

< q 22 (r 2 + s 2 )- *, 
which, by the preceding result, is finite when yu,> 1. Hence 

22 (mco + m () }~-- 

converges uniformly and unconditionally when //, > 1 ; and therefore the least 
value of s, an integer for which 

22 (mco + m co )~ s 
converges uniformly and unconditionally, is 3. 

The series 22(?tto) + m < )~ 2 has a finite sum, the value of which depends* upon 
the infinite limits for the summation with regard to m and m . This dependence is 
inconvenient and it is therefore excluded in view of our present purpose. 

Ex. Prove in the same manner that the series 



the multiple summation extending over all integers m lt m 2 , ...... , m n between oo and 

+ oo , converges uniformly and unconditionally if 2/j.>n. (Eiseustein.) 

57. Returning now to the construction of the transcendental integral 
function the zeros of which are the various points H, we use the preceding 
result in connection with 50 to form the general primary factor. Since 
s = 3, we have 

s-l 



and therefore the primary factor is 



Moreover, the origin is a simple zero. Hence, denoting the required function 
by a (z), we have 

00 

<r(z) = zU H 

00 -00 

as a transcendental integral function which, since the product converges uni 
formly and unconditionally for all finite values of z, exists and has a finite 
value everywhere in the finite part of the plane; the quantity O denotes 
mco + mV, and the double product is taken for all values of m and of m 
between oo and + oo , simultaneous zero values alone being excluded. 

This function will be called Weierstrass s o-function ; it is of importance 
in the theory of doubly-periodic functions which will be discussed in Chapter 
XL 

* See a paper by the author, Quart. Journ. of Math., vol. xxi, (1886), pp. 261280. 



88 PRIMARY FACTORS [57. 

Ex. If the doubly-infinite series of zeros be the points given by 

Q = m 2 ^ + 2wm&> 2 + 2 o> 3 , 

w i> W 2) W 3 being such complex constants that i2 does not vanish for real values of m and n, 
then the series 

2 2 Q-* 

converges for s = 2. The primary factor is thus 



and the simplest transcendental integral function having the assigned zeros is 



The actual points that are the zeros are the intersections of two infinite systems of 
parabolas. 

58. One more result of a negative character will be adduced in this 
connection. We have dealt with the case in which the system of zeros is a 
singly-infinite arithmetical progression of points along one straight line and 
with the case in which the system of zeros is a doubly-infinite arithmetical 
progression of points along two different straight lines : it is easy to see that 
a uniform transcendental integral function cannot exist with a triply -infinite 
arithmetical progression of points for zeros. 

A triply-infinite arithmetical progression of points would be represented 
by all the possible values of 



for all possible integer values for p 1} p.,, p 3 between oo and + oc , where no 
two of the arguments of the complex constants flj, H 2 , O 3 are equal. Let 

tl r = o) r + i(o r , (r = 1, 2, 3) ; 

then, as will be proved ( 107) in connection with a later proposition, it is 
possible* and possible in an unlimited number of ways to determine 
integers p lt p-2,ps so that, save as to infinitesimal quantities, 

Pi _ _ 2 ___ PS 



all the denominators in which equations differ from zero on account of the 
fact that no two arguments of the three quantities fl 1} H 2 , H a are equal. For 
each such set of determined integers we have 

&.Qi+p&+pto 

zero or infinitesimal, so that the origin is a zero of unlimited multiplicity or, 
in other words, there is a space at the origin containing an unlimited number 
of zeros. In either case the origin is an essential singularity, contrary to 

* Jacobi, Oes. Werke, t. ii, p. 27. 



58.] CLASS OF A FUNCTION 89 

the hypothesis that the only essential singularity is for z oo ; and hence a 
uniform transcendental function cannot exist having a triply-infinite arith 
metical succession of zeros. 

59. In effecting the formation of a transcendental integral function by 
means of its primary factors, it was seen that the expression of the primary 
factor depends upon the values of the integers which make 



a converging series. Moreover, the primary factors are not unique in form, 
because any finite number of terms of the proper form can be added to the 
exponential index in 



and such terms will only the more effectively secure the convergence of the 
infinite product. But there is a lower limit to the removal of terms with the 
highest exponents from the index of the exponential ; for there are, in general, 
minimum values for the integers m 1} m,..., below which these integers can 
not be reduced, if the convergence of the product is to be secured. 

The simplest case, in which the exponential must be retained in the 
primary factor in order to secure the convergence of the infinite product, is 
that discussed in 50, viz., when the integers m l , w 2) ... are equal to one 
another. Let m denote this common value for a given function, and let 
m be the least integer effective for the purpose : the function is then said* 
to be of class m, and the condition that it should be of class m is that the 
integer m be the least integer to make the series 



converge uniformly and unconditionally, the constants a being the zeros of 
the function. 

Thus algebraical polynomials are of class ; the circular functions sin z 
and cos z are of class 1 ; Wcierstrass s o--function, and the Jacobian elliptic 
function sn z are of class 2, and so on : but in .no one of these classes do the 
functions mentioned constitute the whole of the functions of that class. 

60. One or two of the simpler properties of an aggregate of transcen 
dental integral functions of the same class can easily be obtained. 

Let a function f(z), of class n, have a zero of order r at the origin and 

* The French word is genre ; the Italian is genere. Laguerre (see references on p. 92) appears 
to have been the first to discuss the class of transcendental integral functions. 



90 



CLASS-PROPERTIES OF 



[60. 



have !, a 2) ... for its other zeros, arranged in order of increasing moduli. 
Then, by 50, the function /O) can be expressed in the form 



(*)= 






M 1 / \ 8 

where </; (V) denotes the series 2 -f 1 and G(z) must be properly deter 
mined to secure the equality. 

Now the series 



is one which converges uniformly for all values of z that do not coincide with 
one of the points a, that is, with one of the zeros of the original function. 
For the sum of the series of the moduli of its terms is 



1 



Let d be the least of the quantities 



1 



, necessarily non-evanescent be 



cause z does not coincide with any of the points a ; then the sum of the series 

IS 1 



which is a converging series since the function is of class n. Hence the 
series of moduli converges and therefore the original series converges ; let it 
be denoted by S (z), so that 

1 



=2 



We have 





Each step of this process is reversible in all cases in which the original pro- 

f ( z \ 
duct converges; if, therefore, it can be shewn of a function f(z) that -rr4 

takes this form, the function is thereby proved to be of class n. 
If there be no zero at the origin, the term - is absent. 



CO.] TRANSCENDENTAL INTEGRAL JUNCTIONS 91 

If the exponential factor G(z) be a constant so that G (z) is zero, the 
function /(.z) is said to be a simple function of class n. 

61. There are one or two criteria to determine the class of a function : 
the simplest of them is contained in the following proposition, due to 
Laguerre*. 

If, as z tends to the value <x> , a very great value of z can be found for 

f (z\ 
which the limit of z~ n --jr\ , where f (z) is a transcendental, integral function, 

J\ z ) 
tends uniformly to the value zero, then f (z} is of class n. 

Take a circle centre the origin and radius R, equal to this value of \z\\ 
then, by 24, II., the integral 

f (t) dt 



JL/lo! 

SvtJ */(*) 



taken round the circle, is zero when R becomes indefinitely great. But the 
value of the integral is, by the Corollary in 20, 



(t) 6A 

+ 



!_ f<*> J./ _(0 Jfc_ _L y ( 
27ri J V- f(t) t-z 2-n-i < =1 J 



t n f(fi t-Z 2-7TI J t n f(t) t-Z 2lri i= i J t n f(t} t-z 

taken round small circles enclosing the origin, the point z, and the points 
a,i, which are the infinities of the subject of integration; the origin being 
supposed a zero of /(t) of multiplicity r. 

1 f !/ (*) dt ._!/ (*) 

JMOW 



t nf( t }t-Z Z n f(2} 
dt I I 

/ \^ / 

Shr, 



1 fWlf(t 
iriJ "/(0 



L f <0> 1 (Q _^_ <^>(^) r 
SwtJ t n f(t)t-z z n z n +* 

where ^> (^) denotes the integral, algebraical, polynomial 



V " f + j~ i -f ~ if + 

when t is made zero. Hence 

and therefore 

which, by GO, shews that/(V) is of class n. 

* Comptcs Rendus, t. xciv, (1882), p. G36. 



92 CLASS-PROPERTIES OF [61. 

COROLLARY. The product of any finite number of functions of the same 
class n is a function of class not higher than n ; and the class of the product 
of any finite number of functions of different classes is not greater than the 
highest class of the component functions. 

The following are the chief references to memoirs discussing the class of functions : 

Laguerrc, Comptes Rendus, t. xciv, (1882), pp. 160-163, pp. 635638, ib. t. xcv, (1882), 
pp. 828831, ib. t. xcviii, (1884), pp. 7981 ; 

Poincare, Bull, des Sciences Math., t. xi, (1883), pp. 136144 ; 

Cesaro, Comptes Rendm, t. xcix, (1884), pp. 2627, followed (p. 27) by a note by 
Hermite; Giornale di Battaglini, t. xxii, (1884), pp. 191 200; 

Vivanti, Giornale di Battaglini, t. xxii, (1884), pp. 243261, pp. 378380, ib. t. xxiii, 
(1885), pp. 96122, ib. t. xxvi, (1888), pp. 303314 ; 

Hermite, Cours d la faculte des Sciences (4 me ed., 1891), pp. 91 93. 



Ex. 1. The function 



2 
1=1 



where the quantities c are constants, n is a finite integer, and the functions J\ (z) are 
algebraical polynomials, is of class unity. 

Ex. 2. If a simple function be of class %, its derivative is also of class n. 

Ex. 3. Discuss the conditions under which the sum of two functions, each of class n, 
is also of class n. 

Ex. 4. Examine the following test for the class of a function, due to Poincare. 

Let a be any number, no matter how small provided its argument be such that e az 
vanishes when z tends towards infinity. Then / (z) is of class n, if the limit of 



vanish with indefinite increase of z. 

A possible value of a is 2 c i a i ~ n ~ 1 , where C; is a constant of modulus unity. 

Ex. 5. Verify the following test for the class of a function, due to de Sparre*. 

Let X be any positive non-infinitesimal quantity ; then the function / (z) is of class n, 
if the limit, for m = oo , of 

\am n ~ l {\a m + i\-\a m \} 
be not less than X. Thus sin z is of class unity. 

Ex. 6. Let the roots of n + 1 = l be 1, a, a 2 , ...... , a n ; and let f (s) be a function 

of class n. Then forming the product 

n/(a4 

we evidently have an integral function of z n + 1 ; let it be denoted by F(z n + 1 ). The roots of 
* Comptes Rendus, t. cii, (1886), p. 741. 



61.] TRANSCENDENTAL INTEGRAL FUNCTIONS 93 

F(z n+l ) = Q are a^ for i=l, 2, and s = 0, 1, , n\ and therefore, replacing z n + 1 by z, 

the roots ofF(z) = are a?* 1 for i=l, 2, ....... 

Since/ (z) is of class n, the series 



converges uniformly and unconditionally. This series is the sum of the first powers of the 
reciprocals of the roots of F(z}~ 0; hence, according to the definition (p. 89), F(z) is of 
class zero. 

It therefore follows that from, a function of any class a function of class zero with a 
modified variable can be deduced. Conversely, by appropriately modifying the variable of 
a given function of class zero, it is possible to deduce functions of any required class. 

Ex. 7. If all the zeros of the function 

=1 r a n r 
\ 
be real, then all the zeros of its derivative are also real. (Witting.) 



00 I / ~ \ 

U\(l--)e 
=* ^\ W 



CHAPTER VI. 

FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES. 

62. SOME indications regarding the character of a function at an 
essential singularity have already been given. Thus, though the function 

is regular in the vicinity of such a point a, it may, like sn - at the origin, 

% 

have a zero of unlimited multiplicity or an infinity of unlimited multiplicity 
at the point ; and in either case the point is such that there is no factor of 
the form (z a) x which can be associated with the function so as to make the 
point an ordinary point for the modified function. Moreover, even when 
the path of approach to the essential singularity is specified, the value 
acquired is not definite : thus, as z approaches the origin along the axis of x, 

so that its value may be taken to be 1 -f- (4>mK + x), the value of sn - is not 

z 

definite in the limit when m is made infinite. One characteristic of the 
point is the indefiniteness of value of the function there, though in the 
vicinity the function is uniform. 

A brief statement and a proof of this characteristic were given in 33 ; 
the theorem there proved that a uniform analytical function can assume 
any value at an essential singularity may also be proved as follows. The 
essential singularity will be taken at infinity a supposition that will be 
found not to detract from generality. 

Let f(z) be a function having any number of zeros and any number 
of accidental singularities and = oo for its sole essential singularity ; then 
it can be expressed in the form 

/w-88*" 

where G 1 (z) is algebraical or transcendental according as the number of zeros 
is finite or infinite and G 2 (z) is algebraical or transcendental according as 
the number of accidental singularities is finite or infinite. 

If Cr 2 (z) be transcendental, we can omit the generalising factor e (z) . 
Then f(z) has an infinite number of accidental singularities ; each of them 
in the finite part of the plane is of only finite multiplicity and therefore some 
of them must be at infinity. At each such point, the function G 2 (z) vanishes 
and O l (z) does not vanish ; and so f(z) has infinite values for z = oo . 



62.] VALUE AT AN ESSENTIAL SINGULARITY 95 

If G z (2) be algebraical and G l (z) be also algebraical, then the factor e a(z) 
may not be omitted, for its omission would make f(z) an algebraical function. 
Now z = oo is either an ordinary point or an accidental singularity of 

ft <*)/<?.<*); 

hence as g (z} is integral there are infinite values of z which make 



infinite. 

If G.>.(z) be algebraical and G^ (z) be transcendental, the factor e g(z) maybe 
omitted. Let a l5 a 2 ,..., a n be the roots of G 2 (z): then taking 

f(z)= ^- 



we have A r = 

a non-vanishing constant ; and so 



where G n (z) is a transcendental integral function. When 2 = oo , the value 
of G 3 (z)/G.,(z) is zero, but G n (z) is infinite ; hence f(z) has infinite values for 

Z= 00 . 

Similarly it may be shewn, as follows, that/(z) has zero values for = oo . 

In the first of the preceding cases, if G l (z) be transcendental, so that f (z) 
has an infinite number of zeros, then some of them must be at an infinite 
distance; f(z) has a zero value for each such point. And if GI(Z) be 
algebraical, then there are infinite values of z which, not being zeros of 
G 2 (z), make f(z) vanish. 

In the second case, when z is made infinite with such an argument as to 
make the highest term in g(z) a real negative quantity, then f(z) vanishes 
for that infinite value of z. 

In the third case,/(V) vanishes for a zero of G 1 (z) that is at infinity. 

Hence the value of f (z) for z= oo is not definite. If, moreover, there 
be any value neither zero nor infinity, say G, which f(z) cannot acquire 
for z = oo , then 

/(*)-C 

is a function which cannot be zero at infinity and therefore all its zeros are 
in the finite part of the plane : no one of them is an essential singularity, for 
f(z) has only a single value at any point in the finite part of the plane; hence 
they are finite in number and are isolated points. Let H 1 (z) be the alge 
braical polynomial having them for its zeros. The accidental singularities 
of f(z} C are the accidental singularities of f(z) ; hence 



96 FORM OF A FUNCTION NEAR [62. 

where, if G 2 (z) be algebraical, the exponential h(z) must occur, since f(z), 
and therefore f(z) C, is transcendental. The function 

-| sy / \ 

TJ 1 ( ~\ _ 2 \ / 0h (z) 

* \*/ f t ~\ n TT f n \ 

J(z)-L H 1 (z) 

evidently has z= oo for an essential singularity, so that, by the second or 
the third case above, it certainly has an infinite value for z = co , that is, 
f(z) certainly acquires the value G for z= GO . 

Hence the function can acquire any value at an essential singularity. 

63. We now proceed to obtain the character of the expression of a 
function at a point z which, lying in the region of continuity, is in the 
vicinity of an essential singularity b in the finite part of the plane. 

With b as centre describe two circles, so that their circumferences and 
the whole area between them lie entirely within the region of continuity. 
The radius of the inner circle is to be as small as possible consistent with 
this condition; and therefore, as it will be assumed that b is the only 
singularity in its own immediate vicinity, this radius may be made very 
small. 

The ordinary point z of the function may be taken as lying within the 
circular ring-formed part of the region of continuity. At all such points in 
this band, the function is holomorphic ; and therefore, by Laurent s Theorem 
( 28), it can be expanded in a converging series of positive and negative 
integral powers of z b in the form 

+ V-L(Z 6)" 1 + v 2 (z 6)~ 2 + . . ., 
where the coefficients u n are determined by the equation 

u n = 



the integrals being taken positively round the outer circle, and the coefficients 
v n are determined by the equation 



the integrals being taken positively round the inner circle. 

The series of positive powers converges everywhere within the outer circle 
of centre b, and so ( 26) it may be denoted by P (z - b) ; and the function P 
may be either algebraical or transcendental. 

The series of negative powers converges everywhere without the inner 
circle of centre b ; and, since 6 is not an accidental but an essential singularity 
of the function, the series of negative powers contains an infinite number of 



63.] AN ESSENTIAL SINGULARITY 97 

terms. It may be denoted by G I -- rh a series converging for all points 

\z o/ 

in the plane except z = b and vanishing when z b = co. 



Thus 



is the analytical representation of the function in the vicinity of its essential 
singularity b ; the function G is transcendental and converges everywhere in 
tlie plane except at z = b, and the function P, if transcendental, converges 
uniformly and unconditionally for sufficiently small values of | z b \ . 

Had the singularity at b been accidental, the function G would have been 
algebraical. 

COROLLARY I. If the function have any essential singularity other than 
b, it is an essential singularity of P (z b) continued outside the outer circle ; 

but it is not an essential singularity of G ( -- j] , for the latter function 

\z ol 

converges everywhere in the plane outside the inner circle. 

COROLLARY II. Suppose the function has no singularity in the plane 
except at the point b ; then the outer circle can have its radius made infinite. 
In that case, all positive powers except the constant term w disappear: 
and even this term survives only in case the function have a finite value at 
infinity. The expression for the function is 



and the transcendental series converges everywhere outside the infinitesimal 
circle round b, that is, everywhere in the plane except at the point b. Hence 
the function can be represented by 



This special result is deduced by Weierstrass from the earlier investiga 
tions*, as follows. If f(z) be such a function with an essential singularity at 
b, and if we change the independent variable by the relation 

Z/== ^b 

thcn/(V) changes into a function of z , the only essential singularity of which 
is at / = GO . It has no other singularity in the plane ; and the form of the 
function is therefore G (z ), that is, a function having an essential singularity 
at b but no other singularity in the plane is 



* Weierstrass (I.e.), p. 27. 
F. 



98 FORM OF A FUNCTION NEAR [63. 

COROLLARY III. The most general expression of a function having its 
sole essential singularity at b a point in the finite part of the plane and any 
number of accidental singularities is 



G, 

where the zeros of the function are the zeros of GI, the accidental singularities 
of the function are the zeros of G 2 , and the function g in the exponential is a 
function which is finite everyiuhere except at b. 

This can be derived in the same way as before ; or it can be deduced 
from the corresponding theorem relating to transcendental integral functions, 
as above. It would be necessary to construct an integral function G 2 (z ) 
having as its zeros 



and then to replace z by - j ; and G., is algebraical or transcendental, 

Z 

according as the number of zeros is finite or infinite. 
Similarly we obtain the following result : 

COROLLARY IV. A uniform function of z, which has its sole essential 
singularity at b a point in the finite part of the plane and no accidental 
singularities, can be represented in the form of an infinite product of primary 
factors of the form 



\z b 

which converges uniformly and unconditionally everywhere in the plane except 
at z = b. 

The function g ( =] is an integral function of T vanishing when 

J \z-bj z-b 

r vanishes; and k and I are constants. In particular factors, q( T ) 

z - b ^ \z - b) 

may vanish ; and either k or I (but not both k and I) may vanish with or 

without a vanishing exponent q { T ) . 

J \z-bj 

If tt{ be any zero, the corresponding primary factor may evidently be 
expressed in the form 

(z 



,z 

Similarly, for a uniform function of z with its sole essential singularity at b and 
any number of accidental singularities, the product-form is at once derivable 



63.] AN ESSENTIAL SINGULARITY 99 

by applying the result of the present Corollary to the result given in 
Corollary III. 

These results, combined with the results of Chapter V., give the complete 
general theory of uniform functions with only one essential singularity. 

64. We now proceed to the consideration of functions, which have a 
limited number of assigned essential singularities. 

The theorem of 63 gives an expression for the function at any point in 
the band between the two circles there drawn. 

Let c be such a point, which is thus an ordinary point for the function ; 
then in the domain of c, the function is expansible in a form P l (z c). 
This domain may extend to an essential singularity b, or it may be limited 
by a pole d which is nearer to c than b is, or it may be limited by an 
essential singularity / which is nearer to c than b is. In the first case, we 
form a continuation of the function in a direction away from b; in the 
second case, we continue the function by associating with the function 
a factor (z d) n which takes account of the accidental singularity ; in 
the third case, we form a continuation of the function towards f. Taking 
the continuations for successive domains of points in the vicinity of/, we can 
obtain the value of the function for points on two circles that have / for 
their common centre. Using these values, as in 63, to obtain coefficients, 
we ultimately construct a series of positive and negative powers converging 
except at / for the vicinity of/ Different expressions in different parts 
of the plane will thus be obtained, each being valid only in a particular 
portion: the aggregate of all of them is the analytical expression of the 
function for the whole of the region of the plane where the function exists. 

We thus have one mode of representation of the function ; its chief 
advantage is that it indicates the form in the vicinity of any point, though it 
gives no suggestion of the possible modification of character elsewhere. This 
deficiency renders the representation insufficiently precise and complete ; and 
it is therefore necessary to have another mode of representation. 

65. Suppose that the function has n essential singularities a,!, a*,..., a n 
and that it has no other singularity. Let a circle, or any simple closed 
curve, be drawn enclosing them all, every point of the boundary as well 
as the included area (with the exception of the n singularities) lying in 
the region of continuity of the function. 

Let z be any ordinary point in the interior of the circle or curve ; and 
consider the integral , f/+\ 

I *=-***> 

taken round the curve. If we surround z and each of the n singularities by 
small circles with the respective points for centres, then the integral round 

72 



100 FUNCTIONS WITH A LIMITED NUMBER [65. 

the outer curve is equal to the sum of the values of the integral taken round 
the n + l circles. Thus 



and therefore 



The left-hand side of the equation isf(z). 
Evaluating the integrals, we have 



where G r is, as before, a transcendental function of - - vanishing when 

1 

is zero. 

z a r 

Now, of these functions, G r {- -] converges everywhere in the plane 

\& \jbfj 

except at a r : and therefore, as n is finite, 



r =i \z - a 
is a function which converges everywhere in the plane except at the n points 

Clj , . . . , a n . 

Because z = oc is not an essential singularity of f(z), the radius of the 

circle in the integral =. . ! /-- dt may be indefinitely increased. The value 

ZTTI J s t z 

of f(t) tends, with unlimited increase of t, to some determinate value G which 
is not infinite ; hence, as in 24, II., Corollary, the value of the integral is 
C. We therefore have the result that/0) can be expressed in the form 



\z-a, 
or, absorbing the constant C into the functions G and replacing the limitation, 

that the function G r ( } shall vanish for = 0, by the limitation 

\z a r j z a r 

that, for the same value =0, it shall be finite, we have the theorem*: 

z a r 

If a given function f(z) have n singularities a^,..., a n , all of which are in 
the finite part of the plane and are essential singularities, it can be expressed 
in the form 

2G f-M, 

r =i r \z - aj 

* The method of proof, by an integration, is used for brevity : the theorem can be established 
by purely algebraical reasoning. 



65.] OF ESSENTIAL SINGULARITIES 101 

where G r is a transcendental function converging everywhere in the plane 

except at a r and having a determinate finite value g r for - = 0, such 

z ci r 

n 

that 2 g r is the finite value of the given function at infinity. 
r=l 

COROLLARY. If the given function have a singularity at oo , and n singu 
larities in the finite part of the plane, then the function can be expressed in 
the form 



w / 1 \ 
G(z) + SG,( L-J, 

r =i \z-a r r 



where G r is a transcendental or an algebraic polynomial function, according 
as a r is an essential or an accidental singularity : and so also for G (z), accord 
ing to the character of the singularity at infinity. 

66. Any uniform function, which has an essential singularity at z = a, 
can ( 63) be expressed in the form 



for points z in the vicinity of a. Suppose that, for points in this vicinity, 
the function f(z) has no zero ; that it has no accidental singularity ; and 
therefore, among such points z, the function 

1 df(z) 
/(*) dz 

has no pole, and therefore no singularity except that at a which is essential. 
Hence it can be expanded in the form 

G(^+P(z-a\ 



z-a 



where G converges everywhere in the plane except at a, and vanishes for 
= 0. Let 






z a dz 



, / 1 \ 

where 0^ I ^ ^ I converges everywhere in the plane except at a, and vanishes 

for = o. 

z a 

Then c, evidently not an infinite quantity, is an integer. To prove this, 
describe a small circle of radius p round a : then taking z-a = pe 91 so that 

= idd, we have 
z a 

l M(*\ 

dz = P (z a) dz 



102 FUNCTIONS WITH A LIMITED NUMBER [66. 

and therefore 



Now JP(z a)dz is a uniform function : and so is f(z). But a change 
of 6 into 6 + 2-7T does not alter z or any of the functions : thus 

actotr 1 

~~ * i 

and therefore c is an integer. 

67, If the function /(z) have essential singularities a lt ..., a n and no 
others, then it can be expressed in the form 

n /I 

C+ $ 9 J- 

r =i \z-a r 

If there be no zeros for this function f(z) anywhere (except of course such 
as may enter through the indeterminateness at the essential singularities), 
then 



/(*) dz 

has n essential singularities a 1} ..., a n and no other singularities of any kind. 
Hence it can be expressed in the form 



n / 1 \ 
C+ 2 G r (- -), 
r =i \z-a, r l 



where the function G r vanishes with . Let 

z a r 

c r d 

I T~~ 



\js a,./ z a r dz { r \z a r 

where G r I ) is a function of the same kind as G r ( ) . 

\z a r / \z a r j 

Then all the coefficients c r , evidently not infinite quantities, are integers. 
For, let a small circle of radius p be drawn round a r : then, if z a r = pe ei , we 
have 

c r dz 



z a r 



= c r i6, 



and ^ = dP s (z - a r ). 

z a s 

We proceed as before : the expression for the function in the former 
case is changed so that now the sum 2P g (0 a r ) for 5 = !,..., i 1, 
r + 1,..., n is a uniform function; there is no other change. In exactly the 
same way as before, we shew that every one of the coefficients c r is an 
integer. 

Hence it appears that if a given function f(z) have, in the finite part of 



67.] OF ESSENTIAL SINGULARITIES 103 

the plane, n essential singularities a l ,..., a n and no other singularities and if 
it have no zeros anywhere in the plane, then 



f(z) dz 

where all the coefficients c* are integers, and the functions G converge every 
where in the plane except at the essential singularities and Gi vanishes for 

-J-- 0. 



Now, since f(z) has no singularity at oo , we have for very large values of z 



and / W = _ > _ 

Z* 

and therefore, for very large values of z, 



_ 

f(z) dz u z 2 z 3 

Thus there is no constant term in =7-^ ^r-^ , and there is no term in -. But 

/(*) dz z 

the above expression for it gives G as the constant term, which must therefore 

vanish ; and it gives 2c; as the coefficient of - , for -7- < (r< ( - H will begin 

z dz [ \z ftj/ J 

with at least ; thus ^a must therefore also vanish. 

Z" 

Hence for a function f (z) which has no singularity at z= oo and no 
zeros anywhere in the plane and of which the only singularities are the n 
essential singularities at a 1} a 2 ,..., a n , we have 



/ (z) dz i= i z - Oi i= i dz ( \z- a 
where the coefficients a are integers subject to the condition 

n 

2 ct = 0. 

i=l 

If a n = oo , so that 2= GO is an essential singularity in addition to a 2 , a 2 ,..., 
a n _j, there is a term 6r (z) instead of G n ( - ] ; there is no term, that corre- 

\Z C^n/ 
/^ 

spends to - , but there may be a constant G. Writing 





z 



with the condition that G (z) vanishes when z 0, we then have 



- 
= __ g ^ 

i=iz-a t dz ( v /J . 



104 PRODUCT-EXPRESSION OF [67. 

where the coefficients d are integers, but are no longer subject to the 
condition that their sum vanishes. 

Let R* (z} denote the function 



the product extending over the factors associated with the essential sin 
gularities of f(z) that lie in the finite part of the plane; thus R*(z) is a 
rational algebraical rneromorphic function. Since 

1 dR*(z) = 2 d 
R* (z) dz ~ i=\z a,i 
we have 

1 df(z) _ 1 dR*(z) = $ d_(-Q ( * 
f(z) dz R* (z) dz i=\dz\ l \z a^ 

where G n ( - ) is to be replaced by G (z) if a n = <x> , that is, if z oo be an 
\z a n j 

essential singularity off(z). Hence, except as to an undetermined constant 
factor, we have 



t=i 

which is therefore an analytical representation of a function with n essential 
singularities, no accidental singularities, and no zeros: and the rational alge 
braical function R* (z) becomes zero or oo only at the singularities off(z). 

If z = oo be not an essential singularity, then R* (z) for z = oo is equal to 

M 

unity because 2 Cf = 0. 
1=1 

COROLLARY. It is easy to see, from 43, that, if the point a; be only an 
accidental singularity, then a is a negative integer and wj I - ) is zero : so 

\Z Oii/ 

that the polar property at c^ is determined by the occurrence of a factor 
(z a{) Ci solely in the denominator of the rational meromorphic function R* (z). 

And, in general, each of the integral coefficients a is determined from the 
expansion of the function f (z) +f(z) in the vicinity of the singularity 
with which it is associated. 

68. Another form of expression for the function can be obtained from 
the preceding; and it is valid even when the function has zeros not 
absorbed into the essential singularities f. 

Consider a function with one essential singularity, and let a be the 
point ; and suppose that, within a finite circle of centre a (or within a finite 
simple curve which encloses a), there are m simple zeros a, /3,..., X of the 

+ See Guichard, TMorie des points singuliers essentiels, (These, Gauthier-Villars, Paris, 1883), 
especially the first part. 



68.] A FUNCTION 105 

function f(z) m being assumed to be finite, and it being also assumed that 
there are no accidental singularities within the circle. Then, if 

/(*) = (* - ) (z - /3). . .(* -\)F (z\ 

the function F (z) has a for an essential singularity and has no zeros within 
the circle. Hence, for points z within the circle, 



where (?, ( ----- ) converges everywhere in the plane and vanishes with - , 

\z a] z a 

and P(z a) is an integral function converging uniformly and unconditionally 
within the circle ; moreover, c is an integer. Thus 



F(z) = A(z- a)" e Gl 
Let (*-a)(*-/3)...(*-X) = (*-a 

_ ( y _ r, \m 

a) 



then f(z) = (* - dT ffl -- F(z} 

\z u/ 

= A(z- ar+ gi (~}e G > ^ e ^~ a] " z . 
\z ft/ 

Now of this product- expression for/(V) it should be noted: 

(i) That m + c is an integer, finite because m and c are finite : 



<?,- 
(ii) The function e ^ z ~ a can be expressed in the form of a series con 

verging uniformly and unconditionally everywhere, except at z = a, and 

proceeding in powers of - in the form 

z a 



.... 

z a (z af 

It has no zero within the circle considered, for F (z) has no zero. Also g l (- - 1 

\z a/ 

algebraical function of , beginning with unity and containing only 



is an 

z a 



a finite number of terms : hence, multiplying the two series together, we have 

as the product a series proceeding in powers of - in the form 

~" a 



a 



which converges uniformly and unconditionally everywhere outside any small 
circle round a, that is, everywhere except at a. Let this series be denoted by 



106 PRODUCT-EXPRESSION OF [68. 

H I ]; it has an essential singularity at a and its only zeros are the 

\z-aj 

points a, (3,..., X, for the series multiplied by g l ( -) has 

\z ft/ 



no zeros : 



(iii) The function fP (z a) dz is a series of positive powers of z a, 
converging uniformly in the vicinity of a; and therefore Q^(z-d)dz can k e 
expanded in a series of positive integral powers of z a which converges 
in the vicinity of a. Let it be denoted by Q (z a) which, since it is a 
factor of F (z), has no zeros within the circle. 

Hence we have 



/(*) = A (z - aYQ (z - a) H 
where p, is an integer ; H ( - J is a series that converges everywhere except 

at a, is equal to unity when - vanishes, and has as its zeros the (finite) 

z a 

number of zeros assigned to f(z) within a finite circle of centre a ; and 
Q (z a) is a series of positive powers of z a beginning with unity which 
converges but has no zero within the circle. 

The foregoing function f(z) is supposed to have no essential singularity 
except at ft. If, however, a given function have singularities at points 
other than a, then the circle would be taken of radius less than the distance 
of a from the nearest essential singularity. 

Introducing a new function f { (z} defined by the equation 



the value of /[ (z) is Q (z a) within the circle, but it is not determined by 
the foregoing analysis for points without the circle. Moreover, as (z a)* 

and also Hi ] are finite everywhere except possibly at a, it follows 

that essential singularities of f(z) other than a must be essential singu 
larities of fj (z). Also since /i (z) is Q(z a) in the immediate vicinity of a, 
this point is not an essential singularity of /i (z). 

Thus /i (z) is a function of the same kind as f(z) ; it has all the essential 
singularities of f(z) except ft, but it has fewer zeros, on account of the m 

zeros of f(z) possessed by H ( - ] . The foregoing expression for f(z) is 

\Z ft/ 

the one referred to at the beginning of the section. 

If we choose to absorb into / x (z} the factors e \ z ~ a and e? P(z ~ dz , 
which occur in 



(z - $* ffl f Jil ^ (T- a) 

\2 ft/ 



68.] A FUNCTION 107 

an expression that is valid within the circle considered, then we obtain a 
result that is otherwise obvious, by taking 



where now g ( ) is algebraical and has for its zeros all the zeros within 

\Z d/ 

the circle ; yu, is an integer; and/j (z) is a function of the same kind as f(z), 
which now possesses all the essential singularities of f(z} but has zeros fewer 

by the in zeros that are possessed by 



z a 



69. Next, consider a function f(z) with n essential singularities a l} 
a 2 ,..., a n but without accidental singularities; and let it have any number of 
zeros. 

When the zeros are limited in number, they may be taken to be isolated 
points, distinct in position from the essential singularities. 

When the zeros are unlimited in number, then at least one of the 
singularities must be such that an infinite number of the zeros lie within 
a circle of finite radius, described round it as centre and containing no other 
singularity. For if there be not an infinite number in such a vicinity 
of some one point (which can only be an essential singularity, otherwise the 
function would be zero everywhere), then the points are isolated and there 
must be an infinite number at z = oo . If z = oo be an essential singularity, the 
above alternative is satisfied : if not, the function, being continuous save at 
singularities, must be zero at all other parts of the plane. Hence it follows 
that if a uniform function have a finite number of essential singularities and 
an infinite number of zeros, all but a finite number of the zeros lie within 
circles of finite radii described round the essential singularities as centres ; 
at least one of the circles contains an infinite number of the zeros, and some 
of the circles may contain only a finite number of them. 

We divide the whole plane into regions, each containing one but only one 
singularity and containing also the circle round the singularity ; let the 
region containing a { be denoted by Ci, and let the region G n be the part of 
the plane other than G lt (7 2 , ...... , G n _^. 

If the region G 1 contain only a limited number of the zeros, then, by 68, 
we can choose a new function /i (z) such that, if 



the function /j (z) has a v for an ordinary point, has no zeros within the region 
G lt and has a 2 , a 3 , ...... , a n for its essential singularities. 

If the region C l contain an unlimited number of the zeros, then, as in 
Corollaries II. and III. of 63, we construct any transcendental function 



108 GENERAL FORM OF A FUNCTION [69. 



5 x f ) , having a^ for its sole essential singularity and the zeros in GI for 

\z OiJ 

all its zeros. When we introduce a function g : (2), defined by the equation 



the function g : {z) has no zeros in GI and certainly has a 2 , a 3 , ...... , a n for 

essential singularities ; in the absence of the generalising factor of G lt it can 
have Hi for an essential singularity. By G7, the function ~g { (2), defined by 

gi (z) = - cO c e hl ^W , 

has no zero and no accidental singularity, and it has a^ as its sole essential 
singularity : hence, properly choosing c x and hi, we may take 

ft(*)-?i(*)/i(*) 

so that fi (z) does not have aj as an essential singularity, but it has all the 
remaining singularities of ^ (z), and it has no zeros within C^. 
In either case, we have a new function f t (z) given by 



where /^ is an integer, the zeros off(z) that lie in GI are the zeros of GI ; the 
function fi(z) has 2 , s> ...... > a n (but not a^ for its essential singularities, 

and it has the zeros of f(z) in the remaining regions for its zeros. 
Similarly, considering (7 2 , we obtain a function / 2 (z), such that 



where /A. 2 is an integer, G 2 is a transcendental function finite everywhere except 
at a 2 and has for its zeros all the zeros of f t (z) and therefore all the zeros of 
f (z) that lie in G 2 ; then f. 2 (z) possesses all the zeros of f(z) in the regions 
other than GI and C 2 , and has a 3 , a 4 ,..., a n for its essential singularities. 

Proceeding in this manner, we ultimately obtain a function f n (z) which 
has none of the zeros off(z) in any of the n regions GI, C 2 ,..., C n , that is, has 
no zeros in the plane, and it has no essential singularities ; it has no acci 

dental singularities, and therefore f n (z) is a constant. Hence, when we 


substitute, and denote by S* (z} the product II (z a^ 1 , we have 



Z 

as the most general form of a function having n essential singularities, no 
accidental singularities, and any number of zeros. The function S* (z) is a 
rational algebraical function of z, usually meromorphic inform, and it has the 
essential singularities off(z) as its zeros and poles ; and the zeros of f (z) are 
distributed among the functions G t . 

As however the distribution of the zeros by the regions C and therefore 



69.] WITH ESSENTIAL SINGULARITIES 109 

the functions G[ ) are somewhat arbitrary, the above form though general 

\z a] 

is not unique. 

If any one of the singularities, say a m , had been accidental and not 

essential, then in the corresponding form the function G m ( - - ) would be 

\Z d m / 

algebraical arid not transcendental. 

70. A function f(z], which has any finite number of accidental singu 
larities in addition to n assigned essential singularities and any number of 
assigned zeros, can be constructed as follows. 

Let A (z) be the algebraical polynomial which has, for its zeros, the 
accidental singularities of f(z), each in its proper multiplicity. Then the 
product 

/(*)-A(*) 

is a function which has no accidental singularities ; its zeros and its essential 
singularities are the assigned zeros and the assigned essential singularities of 
f (z) and therefore it is included in the form 



n ( 
S*(z)U \0 t 

i=i ( 



where S* (z) is a rational algebraical meromorphic function having the points 
Oi, a.,,..., a n for zeros and poles. The form of the function f(z} is therefore 



A 



)} 

-ail) 



71. A function f (z), which has an unlimited number of accidental singu 
larities in addition to n assigned essential singularities and any number of 
assigned zeros, can be constructed as follows. 

Let the accidental singularities be of, /3 ,.... Construct a function f^ (z), 
having the n essential singularities assigned to f (z}, no accidental singu 
larities, and the series a!, /3 ,. . . of zeros. It will, by 69, be of the form of a 
product of n transcendental functions G n+1 ,..., G. 2n which are such that a 
function G has for its zeros the zeros oif- i (z} lying within a region of the plane, 
divided as in 69 ; and the function G n+ t is associated with the point a t -. 

Thus / (z) = T*-(z) ft G n 

f=i 

where T* (z) is a rational algebraical meromorphic function having its zeros 
and its poles, each of finite multiplicity, at the essential singularities ofy(^). 
Because the accidental singularities of f(z) are the same points and have 
the same multiplicity as the zeros of /i (z), the function / (z) / x (z) has no 
accidental singularities. This new function has all the zeros of f(z), and 
a l ,...,a n are its essential singularities; moreover, it has no accidental singu 
larities. Hence the product f(z)fi (z) can be represented in the form 



110 GENERAL FORM OF A FUNCTION [71. 



and therefore we have 



z-fi 



(f-a) 



as an expression of the function. 

But, as by their distribution through the n selected regions of the plane 
in 69, the zeros can to some extent be arbitrarily associated with the 
functions G l} G*,,..., G n and likewise the accidental singularities can to some 
extent be arbitrarily associated with the functions G n+l , G n+ ,..., G^i, the 
product-expression just obtained, though definite in character, is not unique 
in the detailed form of the functions which occur. 

S* (z) 
The fraction 7**) \ 

is algebraical and rational ; and it vanishes or becomes infinite only at the 
essential singularities a lt a. 2 ,..., a n , being the product of factors of the form 
(z i) m s for i = l, 2,..., n. Let the power (z a^ be absorbed into the 
function G{/G n+ i for each of the n values of i ; no substantial change in the 
transcendental character of Gi and of G n+ i is thereby caused, and we may 
therefore use the same symbol to denote the modified function after the 
absorption. Hence "f" the most general product-expression of a uniform 
function of z which has n essential singularities a l} a*,..., a n , any unlimited 
number of assigned zeros and any unlimited number of assigned accidental 
singularities is 

n ^ 

n 



\z-an 

The resolution of a transcendental function with one essential singularity 
into its primary factors, each of which gives only a single zero of the function, 
has been obtained in 63, Corollary IV. 

We therefore resolve each of the functions G^..., G m into its primary 
factors. Each factor of the first n functions will contain one and only one zero 
of the original functions / (.z) ; and each factor of the second n functions will 
contain one and only one of the poles of f(z). The sole essential singularity 
of each primary factor is one of the essential singularities off(z). Hence we 
have a method of constructing a uniform function with any finite number of 
essential singularities as a uniformly converging product of any number of 
primary factors, each of which has one of the essential singularities as its sole 
essential singularity and either (i) has as its sole zero either one of the zeros 

t Weierstrass, I.e., p. 48. 



71.] WITH ESSENTIAL SINGULARITIES 111 

or one of the accidental singularities of/(V), so that it is of the form 



Z \ a ( . 



or (ii) it has no zero and then it is of the form 

/fe). 

When all the primary factors of the latter form are combined, they constitute 
a generalising factor in exactly the same way as in 52 and in 63, 
Cor. III., except that now the number of essential singularities is not 
limited to unity. 

Two forms of expression of a function with a limited number of essential 
singularities have been obtained : one ( 65) as a sum, the other ( 69) as a 
product, of functions each of which has only one essential singularity. Inter 
mediate expressions, partly product and partly sum, can be derived, e.g. 
expressions of the form 



z c. 



But the pure product-expression is the most general, in that it brings into 
evidence not merely the n essential singularities but also the zeros and the 
accidental singularities, whereas the expression as a sum tacitly requires that 
the function shall have no singularities other than the n which are essential. 

Note. The formation of the various elements, the aggregate of which is the complete 
representation of the function with a limited number of essential singularities, can be 
carried out in the same manner as in 34 ; each element is associated with a particular 
domain, the range of the domain is limited by the nearest singularities, and the aggregate 
of the singularities forms the boundary of the region of continuity. 

To avoid the practical difficulty of the gradual formation of the region of continuity 
by the construction of the successive domains when there is a limited number of 
singularities (and also, if desirable to be considered, of branch-points), Fuchs devised 
a method which simplifies the process. The basis of the method is an appropriate change 
of the independent variable. The result of that change is to divide the plane of the 
modified variable f into two portions, one of which, G 2 , is finite in area and the other of 
which, G l , occupies the rest of the plane; and the boundary, common to G l and G 2 , is a 
circle of finite radius, called the discriminating circle* of the function. In G 2 the 
modified function is holomorphic ; in G^ the function is holomorphic except at f = oo ; 
and all the singularities (and the branch-points, if any) lie on the discriminating circle. 

The theory is given in Fuchs s memoir " Ueber die Darstellung der Functionen com- 

plexer Variabeln, ," Crelle, t. Ixxv, (1872), pp. 176 223. It is corrected in details 

and is amplified in Crelle, t. cvi, (1890), pp. 1 4, and in Crelle, t. cviii, (1891), 
pp. 181192; see also Nekrassoff, Math. Ann., t. xxxviii, (1891), pp. 8290, and 
Anissimoff, Math. Ann., t. xl, (1892), pp. 145148. 

* Fuchs calls it Grenzkreis. 



CHAPTER VII. 

FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES, AND EXPANSION 

IN SERIES OF FUNCTIONS. 

72. IT now remains to consider functions which have an infinite number 
of essential singularities*. It will, in the first place, be assumed that the 
essential singularities are isolated points, that is, that they do not form a 
continuous line, however short, and that they do not constitute a continuous 
area, however small, in the plane. Since their number is unlimited and 
their distance from one another is finite, there must be at least one point in 
the plane (it may be at z = oo ) where there is an infinite aggregate of such 
points. But no special note need be taken of this fact, for the character of an 
essential singularity has not yet entered into question ; the essential singu 
larity at such a point would merely be of a nature different from the essential 
singularity at some other point. 

We take, therefore, an infinite series of quantities a 1} a. 2 , a 3 ,... arranged in 
order of increasing moduli, and such that no two are the same : and so we 
have infinity as the limit of a v when v = <x> . 

Let there be an associated series of uniform functions of z such that 
for all values of i. the function G i ( ) , vanishing with , has a { as its 

\Z - Of/ Z Oi 

* The results in the present chapter are founded, except where other particular references are 
given, upon the researches of Mittag-Leffler and Weierstrass. The most important investigations 
of Mittag-Leffler are contained in a series of short notes, constituting the memoir " Sur la th6orie 
des fonctions uniformes d une variable," Comptes Rendus, t. xciv, (1882), pp. 414, 511, 713, 781, 
938, 1040, 1105, 1163, t. xcv, (1882), p. 335 ; and in a memoir " Sur la representation analytique 
des fonctions monogenes uniformes," Acta Math., t. iv, (1884), pp. 1 79. The investigations of 
Weierstrass referred to are contained in his two memoirs " Ueber einen functionentheoretischen 
Satz des Herrn G. Mittag-Leffler," (1880), and " Zur Functionenlehre," (1880), both included in 
the volume Abhandlungen aus der Functionenlehre, pp. 53 66, 67 101, 102 104. A memoir by 
Hermite " Sur quelques points de la theorie des fonctions," Acta Soc. Fenn., t. xii, pp. 67 94, 
Crelle, t. xci, (1881), pp. 54 78 may be consulted with great advantage. 



72.] 



MITTAG-LEFFLER S THEOREM 



113 



sole singularity; the singularity is essential or accidental according as 
GI is transcendental or algebraical. These functions can be constructed 
by theorems already proved. Then we have the theorem, due to Mittag- 
Le frier: It is always possible to construct a uniform analytical function F (z), 
having no singularities other than a 1} a, a,, ... and such that for each deter 
minate value of v, the difference F (z)-G v ( ) is finite for z = a v and 

\z a v / 

therefore, in the vicinity of a v , is expressible in the form P (z ). 

73. To prove Mittag-Leffler s theorem, we first form subsidiary functions 

F v (z), derived from the functions G as follows. The function G v (- } 

\z aj 

converges everywhere in the plane except at the point ; hence within a 
circle z < a v \ it is a monogenic analytic function of z, and can therefore be 
expanded in a series of positive powers of z which converges uniformly 
within the circle, say 



z-a 



for values of z such that \z\ < a v . If a,, be zero, there is evidently no 
expansion. 

Let e be a positive quantity less than 1, and let e lf e 2 , e 3 , ... be arbitrarily 
chosen positive decreasing quantities, subject to the single condition that 2e 
is a converging series, say of sum A : and let e be a positive quantity inter 
mediate between 1 and e. Let g be the greatest value of ~ f 



z a, 



for 



points on or within the circumference \z\ = e a,|; then, because the series 

00 

2 v^z* is a converging series, we have, by 29, 



or 



Hence, with values of z satisfying the condition \z\^.e a v \, we have, for 
any value of m, 



/j.=m 



Vu Z 



2, q - 

9 mJt 

n = m fc o 



1- 



since e<e . Take the smallest integral value of m such that 

9 



F. 



114 MITTAG-LEFFLER S 

it will be finite and may be denoted by m v : and thus we have 



[73. 



for values of z satisfying the condition \z\^.e a v \. 

We now construct a subsidiary function F v (z) such that, for all values of z, 



then for values of UL which are ^ e aJ, 



Moreover, the function 2 zv^ is finite for all finite values of z so that, if we 
n=o 

take 

.j 



-a 



i 



then 6,,(^) is zero at infinity, because, when 5=00, #(- -)i s finite by 

\z ci v / 

hypothesis. Evidently <f> v (z) is infinite only at z = a v , and its singularity is 
of the same kind as that of G t 



z a, 



74. Now let c be any point in the plane, which is not one of the points 
], a 2 , a s , ...; it is possible to choose a positive quantity p such that no one 
of the points a is included within the circle 



z c 



= p- 



Let a v be the singularity, which is the point nearest to the origin satisfy 
ing the condition > c \ + p ; then, for points within or on the circle, we 

have 

z 

a s 

when s has the values v, v + 1, v + 2, Introducing the subsidiary functions 

F v (z), we have, for such values of z, 



and therefore 



F.(z) 



a finite quantity. It therefore follows that the series 2 F, (z) converges uni- 

8=v 

formly and unconditionally for all values of z which satisfy the condition 



74.] THEOREM 115 

z c\^.p. Moreover, all the functions F l (z), F 2 (z), ..., F r _ l (z] are finite for 
such values of z, because their singularities lie without the circle z c = p ; 
and therefore the series 

S F r (z) 

r=l 

converges uniformly and unconditionally for all points z within or on the 
circle \z c =p, where p is chosen so that the circle encloses none of the 
points a. 

The function, represented by the series, can therefore be expanded in the 
form P (z c), in the domain of the point c. 

If a m denote any one of the points a 1} a 2 , ..., and we take p so small that 
all the points, other than a m , lie without the circle 

I / 

I * U"m P ) 

then, since F m (z) is the only one of the functions F which has a singularity 
at a m , the series 

^{F r (z}}-F m (z) 

converges regularly in the vicinity of a, and therefore it can be expressed in 
the form P (z a m ). Hence 



a 



the difference of F m and G m being absorbed into the series P to make Pj. It 

GO 

thus appears that the series 2 F r (z) is a function which has infinities only 

r = \ 

at the points a 1} a 2 , ..., and is such that 



can be expressed in the vicinity of a m in the form P (z - a m ). Hence 2 F r (z) 
is a function of the required kind. 

75. It may be remarked that the function is by no means unique. As 
the positive quantities e were subjected to merely the single condition that 
they form a converging series, there is the possibility of wide variation in 
their choice: and a difference of choice might easily lead to a difference 
in the ultimate expression of the function. 

This latitude of ultimate expression is not, however, entirely unlimited. 
For, suppose there are two functions F(z) and F (z\ enjoying all the assigned 
properties. Then as any point c, other than a^, a 2 , ..., is an ordinary point for 
both F (z) and F (z), it is an ordinary point for their difference : and so 

F(z)-F(z) = P(z-c) 

82 



116 FUNCTIONS POSSESSING [75. 

for points in the immediate vicinity of c. The points a are, however, 
singularities for each of the functions : in the vicinity of such a point a* 
we have 



since the functions are of the required form : hence 

F(z}-F(z}=P(z-a i ) -P(z- ai), 

or the point a; is an ordinary point for the difference of the functions. Hence 
every finite point in the plane, whether an ordinary point or a singularity 
for each of the functions, is an ordinary point for the difference of the 
functions : and therefore that difference is a uniform integral function of z. 
It thus appears that, if F (z) be a function with the required properties, then 
every other function with those properties is of the form 

F(z) + G(z], 

where G (z) is a uniform integral function of z either transcendental or 
algebraical. 

The converse of this theorem is also true. 

00 

Moreover, the function G (z) can always be expressed in a form 2 g v (z), if 

v=\ 

it be desirable to do so : and therefore it follows that any function with the 
assigned characteristics can be expressed in the form 



76. The following applications, due to Weierstrass, can be made so as to 
give a new expression for functions, already considered in Chapter VI., having 
z = oo as their sole essential singularity and an unlimited number of poles at 
points Oi, a 2 , 

If the pole at a f be of multiplicity m i} then (z a$ n >f(z) is regular at 
the point a; and can therefore be expressed in the form 



mi 1 

Hence, if we take / f (z) = 2 c^ (z a i )~ TO < + t , 

M = 

we have f(z} =fi (z) + P (z ;). 

Now deduce from fi(z) a function Fi(z) as in | 73, and let this deduction be 

effected for each of the functions /,- (z). Then we know that 



is a uniform function of z having the points a 1} a 2 , ... for poles in the proper 



76.] UNLIMITED SINGULARITIES 117 

multiplicity and no essential singularity except z = oo . The most general 
form of the function therefore is 



r=\ 

Hence any uniform analytical function which has no essential singularity 
except at infinity can be expressed as a sum of functions each of which has only 
one singularity in the finite part of the plane. The form of F r (z) is 

f r (z}-G r (z\ 

where f r (z) is infinite at z = a r and G r (z) is a properly chosen integral 
function. 

We pass to the case of a function having a single essential singularity at 
c and at no other point and any number of accidental singularities, by taking 

z = - as in 63. Cor. II.: and so we obtain the theorem : 

z c 

Any uniform function which has only one essential singularity, which is 
at c, can be expressed as a sum of uniform functions each of which has only 
one singularity different from c. 

Evidently the typical summative function F r (z) for the present case is of 
the form 



Z 



77. The results, which have been obtained for functions possessed of 
an infinitude of singularities, are valid on the supposition, stated in 72, 
that the limit of a v with indefinite increase of v is infinite ; the series 
ttj, 2 , tends to one definite limiting point which is 2=00 and, by the 
substitution z (z c) = 1, can be made any point c in the finite part of the 
plane. 

Such a series, however, does not necessarily tend to one definite limiting 
point: it may, for instance, tend to condensation on a curve, though the 
condensation does not imply that all points of the continuous arc of the curve 
must be included in the series. We shall not enter into the discussion 
of the most general case, but shall consider that case in which the series of 
moduli \a l) a 2 , ... tends to one definite limiting value so that, with in 
definite increase of v, the limit of \a v is finite and equal to R ; the points 
i, 2 , ... tend to condense on the circle \z = R. 
Such a series is given by 

2fori 
( I _ l -m+n 

,*={! + 



for &=0, 1, ..., n, and n=l, 2, ... ad inf.; and another* by 

aHl + (-l) n c n }e 2M7 " V2 , 
where c is a positive proper fraction. 

* The first of these examples is given by Mittag-Leffler, Acta Math., t. iv, p. 11 ; the second 
was stated to me by Mr Burnside. 



118 FUNCTIONS POSSESSING [77. 

With each point a m we associate the point on the circumference of the 
circle, say b m , to which a m is nearest: let 

| dm "m I = Pm> 

so that p m approaches the limit zero with indefinite increase of m. There 
cannot be an infinitude of points a p , such that p p ^<&, any assigned positive 
quantity ; for then either there would be an infinitude of points a within or 
on the circle \z\ = R , or there would be an infinitude of points a within 
or on the circle z = R + , both of which are contrary to the hypothesis 
that, with indefinite increase of v, the limit of \a v is R. Hence it follows 
that a finite integer n exists for every assigned positive quantity , such that 

\a m -b m \ < 
when m^n. 

Then the theorem, which corresponds to Mittag-LefHer s as stated in 72 
and which also is due to him, is as follows : 

It is always possible to construct a uniform analytical function of z which 
exists over the whole plane, except at the points a and b, and which, in the 
immediate vicinity of each one of the singularities a, can be expressed in the form 



where the functions G{ are assigned functions, vanishing with - - and finite 

Z (Li 

everywhere in the plane except at the single points a; with which they are 
respectively associated. 

In establishing this theorem, we shall need a positive quantity e less than 
unity and a converging series e^ e 2 , e 3 , ... of positive quantities, all less than 
unity. 

Let the expression of the function G n be 

"I / _. .. \0 I / - _. \5 



n \z - a ~ z-a n (z- a n ) 2 (z - a n ) s 

Then, since z - a n = (z - b n ) \l -- ~ l \ , 

( z o n ) 

the function G n can be expressed* in the form 



l <li 

for values of z such that 



a n - 



z-b n 
and the coefficients A are given by the equations 



* The justification of this statement is to be found in the proposition in 82. 



77.] 



UNLIMITED SINGULARITIES 



119 



Now, because G n is finite everywhere in the plane except at a n , the series 



has a finite value, say #, for any non-zero value of the positive quantity % n ; 
then 



Hence 



0*-!)! 



ft & f 

< S flfr-^ 



71 ?^ 

Introducing a positive quantity a such that 



we choose n so that n < a|a n - b n \ ; 

and then | A n> ^ \ < go. ( 1 + a)*- 1 . 

Because (1 + a) e is less than unity, a quantity 6 exists such that 

(1 + a) e < 6 < 1. 



Then for values of z determined by the condition 

go. 6 



d n o n 



< e, we have 



al-0 



Let the integer m n be chosen so that 

ga &> 



it will be a finite integer, because 0< 1. Then 

00 (1 7) 

V I A I "H ^^ 



We now construct, as in 73, a subsidiary function F n (z), defining it by 
the equation 



so that for points z determined by the condition 

\F n (z)\<e n . 
A function with the required properties is 

00 

F m (z\ 



< , we have 



m=l 



120 FUNCTIONS POSSESSING [77. 

To prove it, let c be any point in the plane distinct from any of the points 
a and b ; we can always find a value of p such that the circle 

\z-c\=p 

contains none of the points a and b. Let I be the shortest distance between 
this circle and the circle of radius R, on which all the points b lie ; then for 



all points z within or on the circle 



z c 



p we have 



Now we have seen that, for any assigned positive quantity <s), there is a 
finite integer n such that 

I dm b m < 

when m ^ n. Taking = el, we have 

m 

< e 



when m^n,n being the finite integer associated with the positive quantity el. 
It therefore follows that, for points z within or on the circle \z c\ = p, 

\F m (z}\<e m , 
when m is not less than the finite integer n. Hence 



a finite quantity because e 1} e 2 , ... is a converging series; and therefore 



is a converging series. Each of the functions F 1 (z), F(z), ..., F n _- i (z) is 
finite when z c ^ p ; and therefore 



is a series which converges uniformly and unconditionally for all values of z 
included in the region 

\z-c\^p. 

Hence the function represented by the series can be expressed in the form 
P (z c) for all such values of z. The function therefore exists over the 
whole plane except at the points a and b. 

It may be proved, exactly as in 74, that, for points z in the immediate 
vicinity of a singularity a m , 



The theorem is thus completely established. 

The function thus obtained is not unique, for a wide variation of choice of 
the converging series e a + e 2 + . . . is possible. But, in the same way as in the 



77.] 



UNLIMITED SINGULARITIES 



121 



corresponding case in 75, it is proved that, if F (z) be a function with the 
required properties, every other function with those properties is of the form 

F(z}+G(z\ 

where G (z) behaves regularly in the immediate vicinity of every point in the 
plane except the points b. 

78. The theorem just given regards the function in the light of an 
infinite converging series of functions of the variable : it is natural to suppose 
that a corresponding theorem holds when the function is expressed as an 
infinite converging product. With the same series of singularities as in 
77, when the limit of a v with indefinite increase of v is finite and 
equal to R, the theorem* is: 

It is always possible to construct a uniform analytical function which 
behaves regularly everywhere in the plane except at the points a and b and 
which in the vicinity of any one of the points a v can be expressed in the 
form 



where the numbers w 1} n 2 , ... are any assigned integers. 

The proof is similar in details to proofs of other propositions and it will 
therefore be given only in outline. We have 



a u - 



provided 



such values of z, 



z-a v z-b v z - b v ^i V z - b v J 
< e, the notation being the same as in 77. Hence, for 



=e 



(/7 _ 7) \ 
i_ ^ _ M 
2-bJ 



-n,, S 



by E v (z), we have E v (z} =e m " 

Hence, if F(z) denote the infinite product 






we have F(z) = e 

and F(z) is a determinate function provided the double series in the index of 

the exponential converge. 

* Mittag-Leffler, Acta Math., t. iv, p. 32 ; it may be compared with Weierstrass s theorem in 
67. 



122 TRANSCENDENTAL FUNCTION AS 

Because n v is a finite integer and because 



[78. 



is a converging series, it is possible to choose an integer m v so that 

7) 

"x 



M(T^ 



where t] v is any assigned positive quantity. We take a converging series of 
positive quantities rj v : and then the moduli of the terms in the double series 
form a converging series. The double series itself therefore converges 
uniformly and unconditionally ; and then the infinite product F (z) converges 
uniformly and unconditionally for points z such that 



& b.. 



< e. 



As in 77, let c be any point in the plane, distinct from any of the 
points a and b. We take a finite value of p such that the circle z c\=p 
contains none of the points a and b ; and then, for all points within or on this 
circle, 



z 



<e 



when m^n, n being the finite integer associated with the positive quantity 
el. The product 

fi E v (z) 

v=n 

is therefore finite, for its modulus is less than 

CO 

S IJ K 

K = 



the product 



n 

v=l 



is finite, because the circle z c\ = p contains none of the points a and 6; 
and therefore the function F(z) is finite for all points within or on the circle. 
Hence in the vicinity of c, the function can be expanded in the form P (z c) ; 
and therefore the function exists everywhere in the plane except at the points 
a and b. 

The infinite product converges ; it can be zero only at points which make 
one of the factors zero and, from the form of the factors, this can take place 
only at the points a v with positive integers n v . In the vicinity of a v all 
the factors of F (z) except E v (z) are regular ; hence F (z)\E v (z) can be 
expressed as a function of z a v in the vicinity. But the function has no 
zeros there, and therefore the form of the function is 

P l (z-a,,). 



78.] AN INFINITE SERIES OF FUNCTIONS 123 

Hence in the vicinity of a v , we have 



on combining with P l (z a v ) the exponential index in E v (z). This is the 
required property. 

Other general theorems will be found in Mittag-Leffler s memoir just 
quoted. 

79. The investigations in 72 75 have led to the construction of a 
function with assigned properties. It is important to be able to change, into 
the chosen form, the expression of a given function, having an infinite series 
of singularities tending to a definite limiting point, say to z = oo . It is 
necessary for this purpose to determine (i) the functions F r (z) so that the 

00 

series 2 F r (z) may converge uniformly and (ii) the function G (z). 

r=l 

Let <& (z) be the given function, and let S be a simple contour embracing 
the origin and /j, of the singularities, viz., a l , ...... , a M : then, if t be any 

point, we have 

- . 



m r *) ,,y r *y) ,,. 

J t-z\t) J t-z\t) 



f(a) _ 

where I implies an integral taken round a very small circle centre a. 

If the origin be one of the points a 1} a 2 , ...... , then the first term will be 

included in the summation. 

Assuming that z is neither the origin nor any one of the points a 1} ..., a^, 
we have 



so 



27TI 



AT ^ 

Now . 7-^-7 dt 



1 [ (0) $>(t)fz\ 
. 7-^-7 
Ziri] t-z\tj 



, 
-. 2 I 7^- - dt. 

t-Z\t) 



(ffl-l)i I 

\~d m 1 ((t) + ^ i ^ + ^ 



[ 



124 TRANSCENDENTAL FUNCTION AS [79. 

unless z = be a singularity and then there will be no term G (z). Similarly, 
it can be shewn that 



/ I \ m-l / z \ A 

is equal to G v (- -} - 2 vj-} = F, (z), 

\z - aj A=0 \aj 



where , s 

2?rt 

and the subtractive sum of m terms is the sum of the first m terms in the 
development of G v in ascending powers of z. Hence 



If, for an infinitely large contour, m can be chosen so that the integral 



t- 



diminishes indefinitely with increasing contours enclosing successive singu 
larities, then 



The integer m may be called the critical integer. 
If the origin be a singularity, we take 



and there is then no term G (z) : hence, including the origin in the summa 
tion, we then have 



so that if, for this case also, there be some finite value of m which makes 
the integral vanish, then 



Other expressions can be obtained by choosing for m a value greater than 
the critical integer ; but it is usually most advantageous to take m equal to 
its least lawful value. 

Ex. 1. The singularities of the function ?r cot 772 are given by z = \, for all integer 
values of X from oo to +00 including zero, so that the origin is a singularity. 

The integral to be considered is 



- 1 M IT cot vt fz\ m ,, 
= ~ . I - (- ) at. 

2iri J t-z \tj 



We take the contour to be a circle of very large radius R chosen so that the circumference 
does not pass innuitesimally near any one of the singularities of TT coint at infinity; this 



79.] AN INFINITE SERIES OF FUNCTIONS 125 

is, of course, possible because there is a finite distance between any two of them. Then, 
round the circumference so taken, n cot nt is never infinite : hence its modulus is never 
greater than some finite quantity M. 

Let t = Re ei , so that ~=id6; then 

v 



and therefore 



Z 



.--. 
t-z 



for some point t on the circle. Now, as the circle is very large, we have \t-z\ infinite : 
hence \J\ can be made zero merely by taking m unity. 

Thus, for the function TT cot TTZ, the critical integer is unity. 
Hence from the general theorem we have the equation 



1 fir cot nt z j 

7T COt 772= -5 . 2 I -dt, 

2TTI J t-Z t 



the summation extending to all the points X for integer values of X = - oc to + oo , and 
each integral being taken round a small circle centre X. 

-vr . . 1 /"(*) TT cot irt z , 

Now if, in - . -dt. 

2m J t - z t 

we take t=\ + (, we have 



where P(Q = when = 0; and therefore the value of the integral is 




*./ (*-*+{) (x+fl t 

In the limit when |f| is infinitesimal, this integral 

z 

= (X-2)X 

1 1 

~X-2 X 

and therefore /*. (z) = -J + 1 

A z-X X 

if X be not zero. 

And for the zero of X, the value of the integral is 



( p 



126 REGION OF CONTINUITY [79. 

so that F (z) is -. In fact, in the notation of 72, we have 

z 



o P-AJL 

^ \z-\J~z-\ 
arid the expansion of G K needs to be carried only to one term. 

1 A=ao /I 1\ 
We thus have 7rcot7rs = f- 2 N+=r)> 

z A=-co \Z-X A/ 

the summation not including the zero value of X. 
Ex. 2. Obtain, ab initio, the relation 



SHI 2 3 A = _aj (z-X7r) 2 

p. 3. Shew that, if 



1 1 1 

then "-^^ = - + 2z 2 ^3-^1- 

R(z) z i =l R(\)z*-\* 

(Gylden, Mittag-Leffler.) 

Ex. 4. Obtain an expression, in the form of a sum, for 

IT cot irz 



where Q(z) denotes (1 -z) (l -^ (l -|J ...... ^-j)*- 



80. The results obtained in the present chapter relating to functions 
which have an unlimited number of singularities, whether distributed over 
the whole plane or distributed over only a finite portion of it, shew that 
analytical functions can be represented, not merely as infinite converging 
series of powers of the variable, but also as infinite converging series of 
functions of the variable. The properties of functions when represented by 
series of powers of the variable depended in their proof on the condition that 
the series proceeded in powers; and it is therefore necessary at least to 
revise those properties in the case of functions when represented as series 
of functions of the variable. 

Let there be a series of uniform functions /i (z), /, (z), . . . ; then the 
aggregate of values of z, for which the series 



1*1 

has a finite value, is the region of continuity of the series. If a positive 
quantity p can be determined such that, for all points z within the circle 

z a\ = p, 



80.] OF A SERIES OF FUNCTIONS 127 

00 

the series 2 fi(z) converges uniformly and unconditionally*, the series is 

said to converge in the vicinity of a. If R be the greatest value of p for 
which this holds, then the area within the circle 

z a\ = R 

is called the domain of a; and the series converges uniformly and uncon 
ditionally in the vicinity of any point in the domain of a. 

It will be proved in 82 that the function can be represented by power- 
series, each such series being equivalent to the function within the domain of 
some one point. In order to be able to obtain all the power-series, it is 
necessary to distribute the region of continuity of the function into domains 
of points where it has a uniform, finite value. We therefore form the domain 
of a point 6 in the domain of a from a knowledge of the singularities of the 
function, then the domain of a point c in the domain of 6, and so on ; the 
aggregate of these domains is a continuous part of the plane which has 
isolated points and which has one or several lines for its boundaries. Let 
this part be denoted by A t . 

For most of the functions, which have already been considered, the region 
A 1} thus obtained, is the complete region of continuity. But examples will 
be adduced almost immediately to shew that A-^ does not necessarily include 
all the region of continuity of the series under consideration. Let a be a 
point not in A-^ within whose vicinity the function has a uniform, finite 
value ; then a second portion A 2 can be separated from the whole plane, by 
proceeding from a as before from a. The limits of A and A 2 may be wholly 
or partially the same, or may be independent of one another : but no point 
within either can belong to the other. If there be points in the region of con 
tinuity which belong to neither A 1 nor A 2 , then there must be at least another 
part of the plane A 3 with properties similar to A t and^l 2 - And so on. The 

00 

series 2 fi(z) converges uniformly and unconditionally in the vicinity of 

=i 

every point in each of the separate portions of its region of continuity. 

It was proved that a function represented by a series of powers has a 
definite finite derivative at every point lying actually within the circle 
of convergence of the series, but that this result cannot be affirmed for a 
point on the boundary of the circle of convergence even though the value of 
the series itself should be finite at the point, an illustration being provided 
by the hypergeometric series at a point on the circumference of its circle of 

* In connection with most of the investigations in the remainder of this chapter, Weierstrass s 
memoir " Zur Functionenlehre " already quoted (p. 112, note) should be consulted. 

It may be convenient to give here Weierstrass s definition (I.e., p. 70) of uniform, unconditional 



convergence. A series 2 f n converges uniformly, if an integer m can be determined so that 



/ 



can be made less than any arbitrary positive quantity, however small ; and it converges uncon 
ditionally, if the uniform convergence of the series be independent of any special arrangement 
of order or combination of the terms. 



128 REGION OF CONTINUITY OF [80. 

convergence. It will appear that a function represented by a series of 
functions has a definite finite derivative at every point lying actually within 
its region of continuity, but that the result cannot be affirmed for a point 
on the boundary; and an example will be given ( 83) in which the derivative 
is indefinite. 

Again, it has been seen that a function, initially defined by a given power- 
series, is, in most cases, represented by different analytical expressions in 
different parts of the plane, each of the elements being a valid expression of 
the function within a certain region. The questions arise whether a given 
analytical expression, either a series of powers or a series of functions : 
(i) can represent different functions in the same continuous part of its region 
of continuity, (ii) can represent different functions in distinct (that is, non- 
continuous) parts of its region of continuity. 

81. Consider first a function defined by a given series of powers. 

Let there be a region A in the plane and let the region of continuity of 
the function, say g (z), have parts common with A . Then if a be any point 
in one of these common parts, we can express g {z) in the form P (z a ) in 
the domain of a . 

As already explained, the function can be continued from the domain of 
a by a series of elements, so that the whole region of continuity is gradually 
covered by domains of successive points ; to find the value in the domain of 
any point a, it is sufficient to know any one element, say, the element in the 
domain of a . The function is the same through its region of continuity. 

Two distinct cases may occur in the continuations. 

First, it may happen that the region of continuity of the function g (z) 
extends beyond A . Then we can obtain elements for points outside A , 
their aggregate being a uniform analytical function. The aggregate of 
elements then represents within A a single analytical function : but as that 
function has elements for points without A, the aggregate within A does 
not completely represent the function. Hence 

If a function be defined within a continuous region of a plane by an 
aggregate of elements in the form of power-series, which are continuations of 
one another, the aggregate represents in that part of the plane one (and only 
one) analytical function : but if the power-series can be continued beyond the 
boundary of the region, the aggregate of elements within the region is not the 
complete representation of the analytical function. 

This is the more common case, so that examples need not be given. 

Secondly, it may happen that the region of continuity of the function does 
not extend beyond A in any direction. There are then no elements of the 
function for points outside A and the function cannot be continued beyond 
the boundary of A. The aggregate of elements is then the complete 
representation of the function and therefore : 



81.] A SERIES OF POWERS 129 

If a function be defined within a continuous region of a plane by an 
aggregate of elements in the form of power-series, which are continuations of 
one another, and if the power-series cannot be continued across the boundary of 
that region, the aggregate of elements in the region is the complete representa 
tion of a single uniform monogenic function which exists only for values of the 
variable within the region. 

The boundary of the region of continuity of the function is, in the latter 
case, called the natural limit of the function*, as it is a line beyond which 
the function cannot be continued. Such a line arises for the series 

l + 2z + ^ + 2z 9 + ... , 

in the circle \z = 1, a remark due to Kronecker; other illustrations occur in 
connection with the modular functions, the axis of real variables being the 
natural limit, and in connection with the automorphic functions (see Chapter 
XXII.) when the fundamental circle is the natural limit. A few examples 
will be given at the end of the present Chapter. 

It appears that Weierstrass was the first to announce the existence of natural limits 
for analytic functions, Berlin Monatsber. (1866), p. 617 ; see also Schwarz, Ges. Werke, 
t. ii, pp. 240 242, who adduces other illustrations and gives some references ; Klein and 
Fricke, Vorl. uber die Theorie der elliptischen Modulfunctioncn, t. i, (1890), p. 110; Jordan, 
Cows d Analyse, t. iii, pp. 609, 610. Some interesting examples and discussions of 
functions, which have the axis of real variables for a natural limit, are given by Hankel, 
" Untersuchungen liber die unendlich oft oscillirenden und unstetigen Functionen," 
Math. Ann., t. xx, (1870), pp. 63112. 

82. Consider next a series of functions of the variable ; let it be 



The region of continuity may be supposed to consist of several distinct parts, 
in the most general case ; let one of them be denoted by A. Take some 
point in A, say the origin, which is either an ordinary point or an isolated 
singularity; and let two concentric circles of radii R and R be drawn in A, 
so that 

R < z =r<R, 

and the space between these circles lies within A. In this space, each term 
of the series is finite and the whole series converges uniformly and uncon 
ditionally. 

Now let fi (z) be expanded in a series of powers of z, which series con 
verges within the space assigned, and in that expansion let ^ be the co- 

oo 

efficient of z* ; then we can prove that 2 i^ is finite and that the series 

( / \ 

s |(sO 

n. (\i = I 

* Die natiirliche Grenze, according to German mathematicians. 
F. 



130 REGION OF CONTINUITY [82. 

converges uniformly and unconditionally within this space, so that 

x. (/ oo 

2 /,(*) = 2 2 



i=l " /A {\i=Q 

00 

Because the infinite series 2 fi (z) converges uniformly and uncon 
ditionally, a number n can be chosen so that 



where & is an arbitrary finite quantity, ultimately made infinitesimal; and 
therefore also 



i=n 

where n > n and is infinite in the limit. Now since the number of terms in 
the series 



is not infinite before the limit, we have 



But the original series converges unconditionally, and therefore k is not less 

n 

than the greatest value of the modulus of 2 fi(z) for points within the 

i=n 

region; hence, by 29, we have 

n 

2 V < AT < i . 
= 

00 

Moreover, A; is not less than the greatest value of the modulus of 2 fi(z) 
in the given region ; and so 

00 

2 i^ < AT *. 

i=n 

Now, by definition, k can be made as small as we desire by choice of n ; hence 

the series 



is a converging series. Let it be denoted by A^. 

n-l oo 

Let 2 r M = A /, 2 ifj, = A M " ; 

then, by the above suppositions, we can always choose n so that 



k being any assignable small quantity. 



82.] 



OF A SERIES OF FUNCTIONS 



131 



When two new quantities r and r 2 are introduced, as in 28, satisfying 
the inequalities 

f-f ^ ly ^ \ iv -- /y ^ 7? 

-il/<^/l < s.|.S|<i./2<.-fl, 

the integer w can be chosen so that 

\Ap \ < kr~* < kr^. 

f- r. 



Then 



and 



so that 



2 

. 

00 

2 



- 



- <k 



M =-oo r r-i r 2 -r 

Hence the series 2 A^ z^ can by choice of n be made to have a 
modulus less than any finite quantity ; and therefore, since 

/u.= oo n 1 

(for there is a finite number of terms in the coefficients on each side, the 
expansions are converging series, and the sum on the right-hand side is a 
finite quantity), it follows that the series 



converges uniformly. 
Finally, we have 



2 . 

fl= 00 



2 ft (*) - 24^ = 2 /< (z) - 
<=i 1=1 



and therefore 



2 

t =n 



r ~ 



which, as k can be diminished indefinitely, can be made less than any finite 

jlX=00 

quantity. Hence the series 2 A^ converges unconditionally, and there- 

fi= 00 

fore we have 



provided 



00 jlt=00 

2 /;(*)= 2 . 

l =l /u= oo 



92 



132 REGION OF CONTINUITY [82. 

When we take into account all the parts of the region of continuity 
of the series, constituted by the sum of the functions, we have similar 
expansions in the form of successive series of powers of z c, converging 
uniformly and unconditionally in the vicinities of the successive points c. 
But, in forming the domains of these points c, the boundary of the region of 
continuity of the function must not be crossed ; and a new series of powers is 
required when the circle of convergence of any one series (lying within the 
region of continuity) is crossed. 

It therefore appears that a converging series of functions of a variable 
can be expressed in the form of series of powers of the variable which 
converge within the parts of the plane where the series of functions 
converges uniformly and unconditionally ; but the equivalence of the two 
expressions is limited to such parts of the plane and cannot be extended 
beyond the boundary of the region of continuity of the series of functions. 

If the region of continuity of a series of functions consist of several parts 
of the plane, then the series of functions can in each part be expressed in 
the form of a set of converging series of powers : but the sets of series of 
powers are not necessarily the same for the different parts, and they are not 
necessarily continuations of one another, regarded as power-series. 

Suppose, then, that the region of continuity of a series of functions 

F(z)=lf i (z) 

i=l 

consists of several parts A 1} A. 2 , Within the part A^ let F (z) be 

represented, as above, by a set of power-series. At every point within A 1} 
the values of F(z) and of its derivatives are each definite and unique ; so 
that, at every point which lies in the regions of convergence of two of the 
power-series, the values which the two power-series, as the equivalents of F (z) 
in their respective regions, furnish for F (z) and for its derivatives must be 
the same. Hence the various power-series, which are the equivalents of F (z) 
in the region A ly are continuations of one another: and they are sufficient to 
determine a uniform monogenic analytic function, say F^ (z}. The functions 
F(z) and F l (z) are equivalent in the region A l ; and therefore, by 81, the 
series of functions represents one and the same function for all points within 
one continuous part of its region of continuity. It may (and frequently does) 
happen that the region of continuity of the analytical function F (z) extends 
beyond A ; and then F-^ (z) can be continued beyond the boundary of A^ by 
a succession of elements. Or it may happen that the region of continuity 
of F l (z) is completely bounded by the boundary of A^ ; and then the function 
cannot be continued across that boundary. In either case, the equivalence 

00 

of F-L(Z) and 2 fi(z) does not extend beyond the boundary of A lt one 



82.] OF A SERIES OF FUNCTIONS 133 

00 

complete and distinct part of the region of continuity of 2 fi(z); and 

i = \ 

therefore, by using the theorem proved in 81, it follows that : 

A series of functions of a variable, which converges within a continuous part 
of the plane of the variable z, is either a partial or a complete representation 
of a single uniform, analytic function of the variable in that part of the plane. 

83. Further, it has just been proved that the converging series of 
functions can, in any of the regions A, be changed into an equivalent 
uniform, analytic function, the equivalence being valid for all points in 
that region, say 

2 /(). 4(4 

i = l 

But for any point within A, the function F l (z) has a uniform finite derivative 

oo 

( 21); and therefore also 2 fi(z) has a uniform finite derivative. The 

i=l 

equivalence of the analytic function and the series of functions has not been 
proved for points on the boundary; even if they are equivalent there, the 
function I\ (z) cannot be proved to have a uniform finite derivative at every 

00 

point on the boundary of A, and therefore it cannot be affirmed that 2 ft (z) 

i=\ 

has, of necessity, a uniform, finite derivative at points on the boundary of A, even 

oo 

though the value of 2 fi(z) be uniform and finite at every point on the 

i=l 

boundary*. 

Ex. In illustration of the inference just obtained, regarding the derivative of a 
function at a point on the boundary of its region of continuity, consider the series 

g(z)= 2 &V", 

n=0 

where b is a positive quantity less than unity, and a is a positive quantity which will be 
taken to be an odd integer. 

For points within and on the circumference of the circle \z =1, the series converges 
uniformly and unconditionally; and for all points without the circle the series diverges. 
It thus defines a function for points within the circle and on the circumference, but not 
for points without the circle. 

Moreover for points actually within the circle the function has a first derivative and 
consequently has any number of derivatives. But it cannot be declared to have a 
derivative for points on the circle: and it will in fact now be proved that, if a certain 
condition be satisfied, the derivative for variations at any point on the circle is not merely 
infinite but that the sign of the infinite value depends upon the direction of the variation, 
so that the function is not monogenic for the circumference t. 

* It should be remarked here, as at the end of 21, that the result in itself does not contravene 
Biemann s definition of a function, according to which ( 8) -^ must have the same value what 
ever be tbe direction of the vanishing quantity dz ; at a point on the boundary of the region 
there are outward directions for which die is not defined. 

t The following investigation is due to Weierstrass, who communicated it to Du Bois-Eeymond : 
see Crclle, t. Ixxix, (1875), pp. 2931. 



134 



A SERIES OF FUNCTIONS 



[83. 



Let z = e ei : then, as the function converges unconditionally for all points along the 
circle, we take 

f(ff)= 2 l n e a " ei , 

71=0 

where 6 is a real variable. Hence 



m-l IV,a n (0 + 4>)*_,, WW l 

= s n H - - 
H=O 1 a n $ J 

/ e a">+> (0 + <f>) i _ e a+0(S 

+ 2 &w + M - -T - 1 1 

=o I 9 J 



assuming m, in the first place, to be any positive integer. To transform the first sum on 
the right-hand side, we take 



and therefore 



p a n (0 + <j>) i _ a"0i 



2 (ab} n 

n=0 

< M 2 1 ^n 8Jn(fr-*) 



if ab>\. Hence, on this hypothesis, we have 

2 (ab) n \ \ =y r i 

*=o ( a"0 J ao - 1 

where 7 is a complex quantity with modulus <1. 

To transform the second sum on the right-hand side, let the integer nearest to a m 
be a m , so that 

7T 

for any value of m : then taking 

we have \tr^-x> %n, 

and cos x is not negative. We choose the quantity <f> so that 



and therefore 



TT a m 

ff) , 


which, by taking m sufficiently large (a is > 1), can be made as small as we please. We 
now have 

a m +"(6 +<i>)i = Q a n iti (1 + o) _ _ / _ j N 

if a be an odd integer, and 

_ 



a m + n Oi _ a n i (x + iram] _ / _ j \<m e a n a;i 
, a"xi 



Hence 

CD / 

and therefore 2 &- + f 

,,=0 i 



_ 



- ( - 1) 



2 6" 



83.] MAY NOT POSSESS A DERIVATIVE 135 

The real part of the series on the right-hand side is 

2 b n {l + cosa n x}; 

n=0 

every term of this is positive and therefore, as the first term is 1 + cos x, the real part 

> 1+cos.r 

>1 
for cos x is not negative ; and it is finite, for it is 

<2 2 b n 

K=0 

2 

< r^6- 

Moreover far < TT x < frr, 

so that -- is positive and >-. Hence 
TT x 6 



where TJ is a finite complex quantity, the real part of which is positive and greater than 
unity. We thus have 



where |y |<l, and the real part of 77 is positive and > 1. 
Proceeding in the same way and taking 

IT a m 

TT+X 

so that % = , 

we find t_LJ _ ( _ iy ( a ^ 

where |y/|<l and the real part of TJ^ a finite complex quantity, is positive and greater 
than unity. 

If now we take ab - 1 > fn-, 

the real parts of - + y -*-= , say of f, 

O 7T (tO 1 

and of |li +yi __L_, sayo f fl , 

are both positive and different from zero. Then, since 



and ~x- = (_!)- (ab) m d , 

/(. 

m being at present any positive integer, we have the right-hand sides essentially different 
quantities, because the real part of the first is of sign opposite to the real part of the second. 

Now let m be indefinitely increased; then $ and x are infinitesimal quantities 
which ultimately vanish ; and the limit of - [/(# + </>)-/(#)] for $ = is a complex infinite 



136 ANALYTICAL EXPRESSION [83. 

quantity with its real part opposite in sign to the real part of the complex infinite quantity 
which is the limit of $ ~^ f r = - If# had a differential coefficient 



A 

these two limits would be equal : hence / (0) has not, for any value of 6, a determinate 
differential coefficient. 

From this result, a remarkable result relating to real functions may be at once derived. 
The real part of / (<9) is 

2 6 n cos(a n <9), 

n=0 

which is a series converging uniformly and unconditionally. The real parts of 

-(-ir (&)-<: 

and of +(-l) am (a6) TO f 1 

are the corresponding magnitudes for the series of real quantities : and they are of opposite 
signs. Hence for no value of 6 has the series 

2 6"cos(a n <9) 

n=0 

a determinate differential coefficient, that is, we can choose an increase < and a decrease ^ 
of 6, both being made as small as we please and ultimately zero, such that the limits of 
the expressions 



-X 

are different from one another, provided a be an odd integer and ab > 1 +|TT. 

The chief interest of the above investigation lies in its application to functions of real 
variables, continuity in the value of which is thus shewn not necessarily to imply the 
existence of a determinate differential coefficient defined in the ordinary way. The 
application is due to Weierstrass, as has already been stated. Further discussions will 
be found in a paper by Wiener, Crelle, t. xc, (1881), pp. 221 252, in a remark by 
Weierstrass, Abh. aus der Functionenlehre, (1886), p. 100, and in a paper by Lerch, Crelle, 
t. ciii, (1888), pp. 126 138, who constructs other examples of continuous functions of 
real variables ; and an example of a continuous function without a derivative is given by 
Schwarz, Ges. Werke, t. ii, pp. 269 274. 

The simplest classes of ordinary functions are characterised by the properties : 
(i) Within some region of the plane of the variable they are uniform, finite and 

continuous : 
(ii) At all points within that region (but not necessarily on its boundary) they have 

a differential coefficient : 

(iii) When the variable is real, the number of maximum values and the number of 
minimum values within any given range is finite. 

The function 2 b n cos (a n Q\ suggested by Weierstrass, possesses the first but not the 

71=0 

second of these properties. Kb pcke (Math. Ann., t. xxix, pp. 123 140) gives an example 
of a function which possesses the first and the second but not the third of these 
properties. 

84. In each of the distinct portions A lt A. 2> ... of the complete region of 
continuity of a series of functions, the series can be represented by a 
monogenic analytic function, the elements of which are converging power- 
series. But the equivalence of the function -series and the monogenic 



84.] REPRESENTING DIFFERENT FUNCTIONS 137 

analytic function for any portion A^ is limited to that region. When the 
monogenic analytic function can be continued from A^ into A z , the continua 
tion is not necessarily the same as the monogenic analytic function which is 

00 

the equivalent of the series 2 fi(z) in A 2 . Hence, if the monogenic analytic 

i = l 

functions for the two portions A^ and A 2 be different, the function-series 
represents different functions in the distinct parts of its region of continuity. 

A simple example will be an effective indication of the actual existence 
of such variety of representation in particular cases ; that, which follows, is 
due to Tannery*. 

Let a, b, c be any three constants ; then the fraction 

a + bcz m 
Y+ bz m 
when m is infinite, is equal to a if z \ < 1, and is equal to c if | z > 1. 

Let m , m 1} m 2> ... be any set of positive integers arranged in ascending 
order and be such that the limit of m n , when n = oo , is infinite. Then, 

since 

a + bcz m * a + bcz m {a + bcz m i a + bcz m 



1 + bz m 1 -f bz m f.i (1 + bz m i I + bz " 
^ m o 

" ~* a) 



the function <f)(z), defined by the equation 

,. a + bcz m ., N S f 0^-^-1-1)^-1 

+ (z} = TT6^ + b (G ~ a) {(I + bz^) (i + 6^ 

converges uniformly and unconditionally to a value a if \ z < 1, awe? converges 
uniformly and unconditionally to a value c if z \ > 1. But it does not con 
verge uniformly and unconditionally if z \ = 1. 

The simplest case occurs when b = 1 and m^ = 2* ; then, denoting the 
function by <f> (z), we have 



a - cz , . ( z z 2 z 4 



that is, the function <f> (z) is equal to a if z < 1, and it is equal to c if 



* It is contained in a letter of Tannery s to Weierstrass, who communicated it to the Berlin 
Academy in 1881, Abh. aus der Functionenlehre, pp. 103, 104. A similar series, which indeed is 
equivalent to the special form of $ (z), was given by Schroder, Schlfim. Zeitschrift, t. xxii, (1876), 
p. 184; and Pringsheim, Math. Ann., t. xxii, (1883), p. 110, remarks that it can be deduced, 
without material modifications, from an expression given by Seidel, Crelle, t. Ixxiii, (1871), 
pp. 297- -299. 



138 LINE OF SINGULARITIES [84. 

When \z =\, the function can have any value whatever. Hence a circle 
of radius unity is a line of singularities, that is, it is a line of discontinuity 
for the series. The circle evidently has the property of dividing the plane 
into two parts such that the analytical expression represents different 
functions in the two parts. 

If we introduce a new variable connected with z by the relation* 

l +z 



then, if = + iy and z = x + iy, we have 

1 rfS. nil 

fc i x y 



so that is positive when \z\< 1, and is negative when \ z \ > 1. If then 



the function %() is equal to a or to c according as the real part of f is 
positive or negative. 

And, generally, if we take a rational function of z and denote the 
modified form of </> (), which will be a sum of rational functions of z, by 
^(z), then <f>i(z) will be equal to a in some parts of the plane and to c 
in other parts of the plane. The boundaries between these parts are lines 
of singular points : and they are constituted by the ^-curves which correspond 
to | = 1. 

85. Now let F(z) and G(z) be two functions of z with any number of 
singularities in the plane : it is possible to construct a function which shall 
be equal to F (z} within a circle centre the origin and to G (z) without the 
circle, the circumference being a line of singularities. For, when we make 
a = 1 and c = in </> (z) of 84, the function 

1 z z* z 4 

00)=- -- + -. r + -: : + -. -^r + . . . 
V/ 10 Z 2 I Z* I Z S I 

is unity for all points within the circle and is zero for all points without it : 

and therefore 

G(z} + {F(z)-G(z)}6(z} 

is a function which has the required property. 

Similarly F 3 (z) + {F, (z) - F, (z)} 6 (z) + {F, (z) - F 3 (z}} 6 ( 



is a function which has the value F l (z) within a circle of radius unity, the value F 2 (z) 
between a circle of radius unity and a concentric circle of radius r greater than unity, and 
the value F 3 (z) without the latter circle. All the singularities of the functions F 1} F 2 , F 3 
are singularities of the function thus represented; and it has, in addition to these, the 
two lines of singularities given by the circles. 

* The significance of a relation of this form will be discussed in Chapter XIX. 



85.] MONOGENIC FUNCTIONALITY 139 



Again, 6 

is a function of s, which is equal to F(z) on the positive side of the axis of y, and is equal 
to G (z) on the negative side of that axis. 

1+2 

Also, if we take e l p\ = ^~i 

where a x and p 1 are real constants, as an equation defining a new variable + iy, we have 
| cos a t + 77 sin aj -p l = p. \23T~2 

so that the two regions of the 2-plane determined by \z\<l and \z\>l correspond to the 
two regions of the {"-plane into which the line cos a : + 77 sin a l p 1 = divides it. Let 

,- a i , 1\ 



so that on the positive side of the line cos a t + 77 sin aj p 1 = the function 6 l is unity and 
on the negative side of that line it is zero. Take any three lines defined by a x , p 1 ; a 2 , p 2 , 
a,, pn respectively ; then 

AJ.A11 (2)\-F/(l) 





is a function which has the value F within 
the triangle, the value - F in three of the 
spaces without it, and the value zero in the 
remaining three spaces without it, as indi 
cated in the figure (fig. 13). 

And for every division of the plane by 
lines, into which a circle can be transformed (3) 
by rational equations, as will be explained 
when conformal representation is discussed (1) / 

hereafter, there is a possibility of represent- Fig. 13. 

ing discontinuous functions, by expressions similar to those just given. 

These examples are sufficient to lead to the following result*, which is 
complementary to the theorem of 82 : 

When the region of continuity of an infinite series of functions consists 
of several distinct parts, the series represents a single function in each part 
but it does not necessarily represent the same function in different parts. 

It thus appears that an analytical expression of given form, which con 
verges uniformly and unconditionally in different parts of the plane separated 
from one another, can represent different functions of the variable in those 
different parts ; and hence the idea of monogenic functionality of a complex 
variable is not coextensive with the idea of functional dependence expressible 
through arithmetical operations, a distinction first established by Weierstrass. 

86. We have seen that an analytic function has not a definite value at 
an essential singularity and that, therefore, every essential singularity is 
excluded from the region of definition of the function. 

* Weierstrass, I.e., p. 90. 



140 SINGULAR LINES [86. 

Again, it has appeared that not merely must single points be on occasion 
excluded from the region of definition but also that functions exist with 
continuous lines of essential singularities which must therefore be excluded. 
One method for the construction of such functions has just been indicated : 
but it is possible to obtain other analytical expressions for functions which 
possess what may be called a singular line. Thus let a function have a 
circle of radius c as a line of essential singularity*; let it have no other 
singularities in the plane and let its zeros be a l} a 2 , a 3 ,..., supposed arranged 
in such order that, if p n e ie " = a n > then 

I Pn C | ^ Pn+i ~ C > 

so that the limit of p n , when n is infinite, is c. 

Let c n = ce ie , a point on the singular circle, corresponding to a n which is 
assumed not to lie on it. Then, proceeding as in Weierstrass s theory in 51, 
if 

.= oo ( _ 

Gz= n 



where g n (z) = - + L_ +... + _ 

Z-C n 2 \Z-CnJ m n - I \ Z - C n 

G (z) is a uniform function, continuous everywhere in the plane except along 
the circumference of the circle which may be a line of essential singularities. 

Special simpler forms can be derived according to the character of the 
series of quantities constituted by | a n - c n . If there be a finite integer m, 

00 

such that 2 a n c n m is a converging series, then in g n (z) only the first 

M = l 

m 1 terms need be retained. 
Ex. Construct the function when 



m being a given positive integer and r a positive quantity. 

Again, the point c n was associated with a n so that they have the same 
argument : but this distribution of points on the circle is not necessary and 
can be made in any manner which satisfies the condition that in the limited 

00 

case just quoted the series 2 a n c n m is a converging series. 

Singular lines of other classes, for example, sectioiis\ in connection with functions 
defined by integrals, arise in connection with analytical functions. They are discussed 
by Painleve, "Sur les lignes singulieres des fonctions analytiques," (These, Gauthier- 
Villars, Paris, 1887). 

Ex. Shew that, if the zeros of a function be the points 

. _b+c (a d) i 

ZT ^ ~7 i 7T \ 5 



* This investigation is due to Picard, Comptes Rendus, t. xci, (1881), pp. 690692. 
t Called conpures by Hermite ; see 103. 



86.] LACUNARY FUNCTIONS 141 

where a, ?;, c, d are integers satisfying the condition ad-bo = l, so that the function 
has a circle of radius unity for an essential singular line, then if 

b + di 





2J = -^ - = , , 

d+bi 



( A 

the function n \ 5 e z 

(z li 

where the product extends to all positive integers subject to the foregoing condition 
ad-bc = l, is a uniform function finite for all points in the plane not lying on the 
circle of radius unity. (Picard.) 

87. In the earlier examples, instances were given of functions which 
have only isolated points for their essential singularities : and, in the later 
examples, instances have been given of functions which have lines of 
essential singularities, that is, there are continuous lines for which the 
functions do not exist. We now proceed to shew how functions can be 
constructed which do not exist in assigned continuous spaces in the plane, 
these spaces being aggregates of essential singularities. Weierstrass was 
the first to draw attention to lacunary functions, as they may be called ; 
the following investigation in illustration of Weierstrass s theorem is due to 
Poincare *. 

Take any convex curve in the plane, say G ; and consider the function 

*z^b 
where the quantities A are constants, subject to the conditions 

(i) The series ^\A\ converges uniformly and unconditionally : 
(ii) Each of the points b is either within or on the curve G : 
(iii) The points b are the aggregate of all rational j points within and 
on C : then the function is a uniform analytical function for all points 
without C and it has the area of G for a lacunary space. 

First, it is evident that, if z = b, then the series does not converge. 
Moreover as the points b are the aggregate of all the rational points within 
or on C, there will be an infinite number of singularities in the immediate 
vicinity of b : we shall thus have an unlimited number of terms each infinite 
of the first order, and thus ( 42) the point b will be an essential singularity. 
As this is true of all points z within or on C, it follows that the area C is a 
lacunary space for the function, if the function exist at all. 

Secondly, let z be a point without G ; and let d be the distance of z from 
the nearest point of the boundary of C^f% so that d is not a vanishing quantity. 

* Acta Soc. Fenn., t. xii, (1883), pp. 341350. 

J Rational points within or on C are points whose positions can be determined rationally in 
terms of the coordinates of assigned points on C ; examples will be given. 

t This will be either the shortest normal from z to the boundary or the distance of z from 
some point of abrupt change of direction, as for instance at the angular point of a polygon. 



142 



FUNCTIONS WITH 



[87. 



Then | z b \ ^ d ; and therefore 

A _ \A\ \A\ 
~\z-b\ < d 



z-b 



so that 



-b 



A 



z-b 



Now 2 j.A| converges uniformly and unconditionally and therefore, as d does 
not vanish, 



z-b 
converges uniformly and unconditionally, that is. 



is a function of 2 which converges uniformly and unconditionally for every 
point without C. Let it be denoted by < (z). 

Let c be any point without C, and let r be the radius of the greatest 
circle centre c which can be drawn so as to have no point of C within itself 
or on its circumference, so that r is the radius of the domain of c; then 
b c > r, for all points b. 

If we take a point z within this circle, we have \z c =6r, where 6 < 1. 

Now for all points within this circle the function < {z} converges uniformly, 

A 

and every term -- = of <f> (z) is finite. Also, for points within the circle, we 

A 

can expand -- j in powers of z c in the form 









of a converging series. Hence, by 82, we have 

<(*)= 2 B m (z-c) m , 



a series converging uniformly and unconditionally for all points within the 
circle centre c and radius r, which circle is the circle of convergence of the 
series. The function can be expressed in the usual manner over the whole of 
the region of continuity, which is the part of the plane without the curve C. 

Thus (z) is a uniform analytical function, having the area of C for a 
lacunary space. 

As an example, take a convex polygon having o 1} ...... , a p for its angular points; 

then any point 

...... +m j >a p 



TOI + ...... +m p 

where m lt ...... , m p are positive integers or zero (simultaneous zeros being excluded), is 



87.] LACUNARY SPACES 143 

either within the polygon or on its boundary : and any rational point within the polygon 

or on its boundary can be represented by 

p 
2 m r a r 

r=l 

P 
2 m r 
r=l 
by proper choice of ?n 15 ...... , m p , a choice which can be made in an infinite number of ways. 

Let u lt ...... , Up be given quantities, the modulus of each of which is less than unity: 

then the series 

9-11 m > 11 m f 

& ^ I ...... ftp 

o 

converges uniformly and unconditionally. Then all the assigned conditions are satisfied 
for the function 



_ .. . + m p a p > 

m l + ...... +m p J 

and therefore it is a function which converges uniformly and unconditionally everywhere 
outside the polygon and which has the polygonal space (including the boundary) for 
a lacunary space. 

If, in particular, p = Z, we obtain a function which has the straight line 
joining a x and a 2 as a line of essential singularity. When we take a t = 0, 
a. 2 = 1 and slightly modify the summation, we obtain the function 



2 2 ^ 2 

w=l m=0 W& 

7i 

which, when u^ <\ and |w 2 |<l, converges uniformly and unconditionally 
everywhere in the plane except at points between and 1 on the axis of real 
quantities, this part of the axis being a line of essential singularity. 
For the general case, the following remarks may be made : 

(i) The quantities u 1} u 2> ... need not be the same for every term; a 
numerator, quite different in form, might be chosen, such as 
(mj 2 + ... + m/)" 1 where 2//, > p ; all that is requisite is that the 
series, made up of the numerators, should converge uniformly 
and unconditionally. 

(ii) The preceding is only a particular illustration and is not necessarily 
the most general form of function having the assigned lacunary 
space. 

It is evident that the first step in the construction of a function, which 
shall have any assigned lacunary space, is the formation of some expression 
which, by the variation of the constants it contains, can be made to 
represent indefinitely nearly any point within or on the contour of the 
space. Thus for the space between two concentric circles of radii a and c 
and centre the origin we should take 

Wja + O-WjU ^a 
- a - e n 

n 



144 EXAMPLES [87. 

which, by giving m^ all values from to n, ra 2 all values from to n 1 and 
n all values from 1 to infinity will represent all rational points in the space : 
and a function, having the space between the circles as lacunary, would be 
given by 



oo n n-1 

2 2 2 

n=l !=(> m 2 =0 



(n raj) b ^ 271- 



r /3 

.6 C 

n 



provided u\ < 1, u^ < 1, u 2 < 1. 

In particular, if a = 6, then the common circumference is a line of essential singularity 
for the corresponding function. It is easy to see that the function 




z ae n 



ao 2n-l m n 

provided the series 2 2 u v 

n=l m=0 m,n m, n 

converges uniformly and unconditionally, is a function having the circle |0| = a as a line of 
essential singularity. 

Other examples will be found in memoirs by Goursat*, Poincaref, and HomenJ. 

Ex. 1. Shew that the function 



where r is a real positive quantity and the summation is for all integers m and n between 
the positive and the negative infinities, is a uniform function in all parts of the plane 
except the axis of real quantities which is a line of essential singularity. 

Ex. 2. Discuss the region in which the function 



w=i m=i jf/=i i ^- . - 
21 1 i 

\7i 71 

is definite. (Homen.) 

Ex. 3. Prove that the function 

n=0 

exists only within a circle of radius unity and centre the origin. (Poincare.) 

Ex. 4. An infinite number of points a t , a 2 , a s , are taken on the circumference of 

a given circle, centre the origin, so that they form the aggregate of rational points on the 
circumference. Shew that the series 

2 l Z 

can be expanded in a series of ascending powers of z which converges for points within the 
circle, but that the function cannot be continued across the circumference of the circle. 

(Stieltjes.) 

* Comptes Rendus, t. xciv, (1882), pp. 715718 ; Bulletin de Darboux, 2 me Ser. , t. xi, (1887), 
pp. 109114. 

t In the memoir, quoted p. 138, and Comptes Rendus, t. xcvi, (1883), pp. 1134-1136. 
+ Acta Soc. Fenn., t. xii, (1883), pp. 445464. 



87.] EXAMPLES 145 

Ex. 5. Prove that the series 



2 | : 

7T .00 -" K1-2TO- 
9 oo 



22 



~ l i) 2 ) 



7T _oo _oo (^(1 2wi 2nz i) \zm-\-Anz~ 

where the summation extends over all positive and negative integral values of ra and of n 
except simultaneous zeros, is a function which converges uniformly and unconditionally 
for all points in the finite part of plane which do not lie on the axis of y ; and that 
it has the value +1 or - 1 according as the real part of z is positive or negative. 

(Weierstrass.) 
Ex. 6. Prove that the region of continuity of the series 



consists of two parts, separated by the circle z\ = l which is a line of infinities for 
the series : and that, in these two parts of the plane, it represents two different 
functions. 

_<a ir 

If two complex quantities a> and to be taken, such that z = e ^ and the real part of 
^. is positive, and if they be associated with the elliptic function $ (u) as its half-periods, 
then for values of z which lie within the circle z = \ 



in the usual notation of Weierstrass s theory of elliptic functions. 

Find the function which the series represents for values of z without the circle \z\ = \. 

(Weierstrass.) 

Ex. 7. Four circles are drawn each of radius -^ having their centres at the points 

1, i, - 1, -i respectively; the two parts of the plane, excluded by the four circumferences, 
are denoted the interior and the exterior parts. Shew that the function 

n = K sini^TT ( 1 1 1 1 



is equal to IT in the interior part and is zero in the exterior part. (Appell.) 

Ex. 8. Obtain the values of the function 

;-l- (-!)(, i >1 l 

=i n V 1 > (2 + l) (2-l)J 

in the two parts of the area within a circle centre the origin and radius 2 which lie 

without two circles of radius unity, having their centres at the points 1 and - 1 

respectively. (Appell.) 



Ex - 9 - If 
and 



,~ 3 ...... 

a m r (2- m ) 3 J 

where the regions of continuity of the functions F extend over the whole plane, then / (z) 
is a function existing everywhere except within the circles of radius unity described round 
the points a, , 2 , ...... , a n . (Teixeira.) 

F - 10 



146 CLASSIFICATION [87. 

Ex. 10. Let there be n circles having the origin for a common centre, and let 
,, (7 2 , ...... , (7 n , C n + 1 be % + 1 arbitrary constants; also let a 1} a 2 , ...... , a n be any w points 

lying respectively on the circumferences of the first, the second, ...... , the nth circles. 

Shew that the expression 



1 ("(CL 
27T./0 W* 



has the value <7 m for points z lying between the (w - l)th and the with circles and the 
value (7 n + 1 for points lying without the nth circle. 

Construct a function which shall have any assigned values in the various bands into 
which the plane is divided by the circles. (Pincherle.) 

88. In 32 it was remarked that the discrimination of the various 
species of essential singularities could be effected by means of the properties 
of the function in the immediate vicinity of the point. 

Now it was proved, in 63, that in the vicinity of an isolated essential 
singularity b the function could be represented by an expression of the form 



for all points in the space without a circle centre b of small radius and within 
a concentric circle of radius not large enough to include singularities at 
a finite distance from b. Because the essential singularity at b is isolated, 
the radius of the inner circle can be diminished to be all but infinitesimal : 

the series P (z b) is then unimportant compared with G I 31 ) , which 
can be regarded as characteristic for the singularity of the function. 

Another method of obtaining a function, which is characteristic of the 
singularity, is provided by 68. It was there proved that, in the vicinity of 
an essential singularity a, the function could be represented by an expression 
of the form 



where, within a circle of centre a and radius not sufficiently large to include 
the nearest singularity at a finite distance from a, the function Q (z a) is 
finite and has no zeros : all the zeros of the given function within this circle 
(except such as are absorbed into the essential singularity at a) are zeros of 

the factor H ( - - ] , and the integer-index n is affected by the number of these 
zeros. When the circle is made small, the function 



z-a 



can be regarded as characteristic of the immediate vicinity of a or, more 
briefly, as characteristic of a. 



88.] OF SINGULARITIES 147 

It is easily seen that the two characteristic functions are distinct. For 
if F and F^ be two functions, which have essential singularities at a of the 
same kind as determined by the first characteristic, then 

F(z)-F l (z) = P(z-a)-P l (z-a) 
= P,(z-a\ 

while if their singularities at a be of the same kind as determined by the 
second characteristic, then 

F(z)_Q(z-a) 
f\(*)-Q^-~a) = Q ^ 2 - 

in the immediate vicinity of a, since Q 1 has no zeros. Two such equations 
cannot subsist simultaneously, except in one instance. 

Without entering into detailed discussion, the results obtained in the 
preceding chapters are sufficient to lead to an indication of the classification 
of singularities*. 

Singularities are said to be of the first class when they are accidental ; 
and a function is said to be of the first class when all its singularities are of 
the first class. It can, by 48, have only a finite number of such singularities, 
each singularity being isolated. 

It is for this case alone that the two characteristic functions are in 
accord. 

When a function, otherwise of the first class, fails to satisfy the last 
condition, solely owing to failure of finiteness of multiplicity at some point, 
say at z = x , then that point ceases to be an accidental singularity. It has 
been called ( 32) an essential singularity ; it belongs to the simplest kind of 
essential singularity ; and it is called a singularity of the second class. 

A function is said to be of the second class when it has some singularities 
of the second class ; it may possess singularities of the first class. By an 
argument similar to that adopted in 48, a function of the second class 
can have only a limited number of singularities of the second class, each 
singularity being isolated. 

When a function, otherwise of the second class, fails to satisfy the last 
condition solely owing to unlimited condensation at some point, say at z = oo , 
of singularities of the second class, that point ceases to be a singularity 
of the second class: it is called a singularity (necessarily essential) of the 
third class. 

* For a detailed discussion, reference should be made to Guichard, " Theorie des points 
singnhers essentiels" (These, Gauthier-ViUars, Paris, 1883), who gives adequate references to the 

:stigations of Mittag-Leffler in the introduction of the classification and to the researches of 
Cantor. See also Mittag-Leffler, Acta Math., t. iv, (1884), pp. 1_79; Cantor Crelle t Ixxxiv 
1878), pp. 242258, Acta Math., t. ii, (1883), pp. 311328. 

102 



148 CLASSIFICATION OF SINGULARITIES [88. 

A function is said to be of the third class when it has some singularities 
of the third class ; it may possess singularities of the first and the second 
classes. But it can have only a limited number of singularities of the third 
class, each singularity being isolated. 

Proceeding in this gradual sequence, we obtain an unlimited number of 
classes of singularities: and functions of the various classes can be constructed 
by means of the theorems which have been proved. A function of class n 
has a limited number of singularities of class n, each singularity being 
isolated, and any number of singularities of lower classes which, except in so 
far as they are absorbed in the singularities of class n, are isolated points. 

The effective limit of this sequence of classes is attained when the 
number of the class increases beyond any integer, however large. When 
once such a limit is attained, we have functions with essential singularities of 
unlimited class, each singularity being isolated ; when we pass to functions 
which have their essential singularities no longer isolated but, as in previous 
class-developments, of infinite condensation, it is necessary to add to the 
arrangement in classes an arrangement in a wider group, say, in species*. 

Calling, then, all the preceding classes of functions functions of the first 
species, we may, after Guichard (I.e.), construct, by the theorems already 
proved, a function which has at the points a l} a*,... singularities of classes 
1, 2,..., both series being continued to infinity. Such a function is called 
a function of the second species. 

By a combination of classes in species, this arrangement can be continued 
indefinitely ; each species will contain an infinitely increasing number of 
classes; and when an unlimited number of species is ultimately obtained, 
another wider group must be introduced. 

This gradual construction, relative to essential singularities, can be carried 
out without limit ; the singularities are the characteristics of the functions. 

* Guichard (I.e.) uses the term genre. 



CHAPTER VIII. 

MULTIFORM FUNCTIONS. 

89. HAVING now discussed some of the more important general properties 
of uniform functions, we proceed to discuss some of the properties of multiform 
functions. 

Deviations from uniformity in character may arise through various causes : 
the most common is the existence of those points in the ^-plane, which have 
already ( 12) been defined as branch-points. 

As an example, consider the two power-series 

Wl = l-i/-i/ 2 -... , W2 = _(i_i/_^_... )f 

which, for points in the plane such that z is less than unity, are the two 
values of (1 - /)* ; they may be regarded as two branches of the function w 
defined by the equation 

w 2 = 1 z = z. 

Let / describe a small curve (say a circle of radius r) round the point 
z = l, beginning on the axis of x\ the point 1 is the origin for z. Then z 
is r initially, and at the end of the first description of the circle z is re 2wi ; 
hence initially w l is + 1 4 and w. 2 is - r* } an d at the end of the description 
w 1 is -f r^e and w 2 is r^e, that is, w l is rf and w. 2 is + ri Thus the 
effect of the single circuit is to change w l into w. 2 and w 2 into w 1} that is, 
the effect of a circuit round the point, at which w 1 and w 2 coincide in value, 
is to interchange the values of the two branches. 

If, however, z describe a circuit which does not include the branch-point, 
w l and w 2 return each to its initial value. 

Instances have already occurred, e.g. integrals of uniform functions, in 
which a variation in the path of the variable has made a difference in the 



150 CONTINUATIONS [89. 

result; but this interchange of value is distinct from any of the effects 
produced by points belonging to the families of critical points which have 
been considered. The critical point is of a new nature ; it is, in fact, a 
characteristic of multiform functions at certain associated points. 

We now proceed to indicate more generally the character of the relation 
of such points to functions affected by them. 

The method of constructing a monogenic analytic function, described in 
34, by forming all the continuations of a power-series, regarded as a given 
initial element of the function, leads to the aggregate of the elements of the 
function and determines its region of continuity. When the process of con 
tinuation has been completely carried out, two distinct cases may occur. 

In the first case, the function is such that any and every path, leading 
from one point a to another point z by the construction of a series of 
successive domains of points along the path, gives a single value at z as the 
continuation of one initial value at a. When, therefore, there is only a 
single value of the function at a, the process of continuation leads to only a 
single value of the function at any other point in the plane. The function is 
uniform throughout its region of continuity. The detailed properties of such 
functions have been considered in the preceding chapters. 

In the second case, the function is such that different paths, leading from 
a to z, do not give a single value at z as the continuation of one and the 
same initial value at a. There are different sets of elements of the function, 
associated with different sets of consecutive domains of points on paths from 
a to z, which lead to different values of the function at z; but any change 
in a path from a to z does not necessarily cause a change in the value of the 
function at z. The function is multiform in its region of continuity. The 
detailed properties of such functions will now be considered. 

90. In order that the process of continuation may be completely carried 
out, continuations must be effected, beginning at the domain of any point a 
and proceeding to the domain of any other point b by all possible paths in 
the region of continuity, and they must be effected for all points a and b. 
Continuations must be effected, beginning in the domain of every point a 
and returning to that domain by all possible closed paths in the region of 
continuity. When they are effected from the domain of one point a to that 
of another point b, all the values at any point z in the domain of a (and not 
merely a single value at such points) must be continued : and similarly when 
they are effected, beginning in the domain of a and returning to that domain. 
The complete region of the plane will then be obtained in which the function 
can be represented by a series of positive integral powers : and the boundary 
of that region will be indicated. 




90.] OF A MULTIFORM FUNCTION 151 

In the first instance, let the boundary of the region be constituted by a 
number, either finite or infinite, of 
isolated points, say L 1} L 2 , L s , ... 
Take any point A in the region, so 
that its distance from any of the 
points L is not infinitesimal ; and 
in the region draw a closed path 
ABC...EFA so as to enclose one 
point, say L l} but only one point, of 
the boundary and to have no point 

of the curve at a merely infinitesimal distance from L^ Let such curves be 
drawn, beginning and ending at A, so that each of them encloses one and 
only one of the points of the boundary : and let K r be the curve which 
encloses the point L r . 

Let Wj be one of the power-series defining the function in a domain with 
its centre at A : let this series be continued along each of the curves K s by 
successive domains of points along the curve returning to A. The result 
of the description of all the curves will be that the series w^ cannot be 
reproduced at A for all the curves though it may be reproduced for some 
of them ; otherwise, w : would be a uniform function. Suppose that w. 2 , w 3> ..., 
each in the form of a power-series, are the aggregate of new distinct values 
thus obtained at A ; let the same process be effected on w 2 , w 3 , ... as has 
been effected on w 1; and let it further be effected on any new distinct values 
obtained at A through w 2 , w 3 , ... , and so on. When the process has 
been carried out so far that all values obtained at A, by continuing any 
series round any of the curves K back to A, are included in values already 
obtained, the aggregate of the values of the function at A is complete : they 
are the values at A of the branches of the function. 

We shall now assume that the number of values thus obtained is finite, 
say n, so that the function has n branches at A : if their values be denoted 
by w 1} w 2 , ..., w n , these n quantities are all the values of the function at A. 
Moreover, n is the same for all points in the plane, as may be seen by con 
tinuing the series at A to any other point and taking account of the corollaries 
at the end of the present section. 

The boundary-points L may be of two kinds. It may (and not infre 
quently does) happen that a point L s is such that, whatever branch is taken 
at A as the initial value for the description of the circuit K s , that branch is 
reproduced at the end of the circuit. Let the aggregate of such points be 
/u J 2 , .... Then each of the remaining points L is such that a description 
of the circuit round it effects a change on at least one of the branches, taken 
as an initial value for the description ; let the aggregate of these points be 
B lt 5 2 , .... They are the branch-points; their association with the definition 
in 12 will be made later. 



152 



DEFORMATION OF PATH 



[90. 




Fig. 15. 



When account is taken of the continuations of the function from a point 
A to another point B, we have n values at B as the continuations of n values 
at A. The selection of the individual branch at B, which is the continuation 
of a particular branch at A, depends upon the path of z between A and 5; 
it is governed by the following fundamental proposition : 

The final value of a branch of a function for two paths of variation of the 
independent variable from one point to another will be the same, if one path 
can be deformed into the other without passing over a branch-point. 

Let the initial and the final points be a and b, and let one path of 
variation be acb. Let another path of variation be aeb, 
both paths lying in the region in which the function can 
be expressed by series of positive integral powers : the two 
paths are assumed to have no point within an infinitesimal 
distance of any of the boundary-points L and to be taken 
so close together that the circles of convergence of pairs of 
points (such as c x and e 1} c 2 and e 2 , and so on) along the two 
paths have common areas. When we begin at a with a 
branch of the function, values at d and at e^ are obtained, 
depending upon the values of the branch and its derivatives at a and upon 
the positions of c a and e^ hence, at any point in the area common to the 
circles of convergence of these two points, only a single value arises as 
derived through the initial value at a. Proceeding in this way, only a single 
value is obtained at any point in an area common to the circles of con 
vergence of points in the two paths. Hence ultimately one and the same 
value will be obtained at b as the continuation of the value of the one branch 
at a by the two different paths of variation which have been taken so that 
no boundary-point L lies between them or infinitesimally near to them. 

Now consider any two paths from a to b, say acb and adb, such that 
neither of them is near a boundary-point and that the 
contour they constitute does not enclose a boundary-point. 
Then by a series of successive infinitesimal deformations we 
can change the path acb to adb ; and as at b the same value 
of w is obtained for variations of z from a to b along the 
successive deformations, it follows that the same value of w 
is obtained at b for variations of z along acb as for varia 
tions along adb. 

Next, let there be two paths acb, adb constituting a closed contour, 
enclosing one (but not more than one) of the points / and none of the points 
B. When the original curve K which contains the point / is described, the 
initial value is restored : and hence the branches of the function obtained at 
any point of K by the two paths from any point, taken as initial point, are 
the same. By what precedes, the parts of this curve K can be deformed 




Fig. 16. 



90.] OF THE VARIABLE 153 

into the parts of acbda without affecting the branches of the function : hence 
the value obtained at b, by continuation along acb, is the same as the value 
there obtained by continuation along adb. It therefore follows that a path 
between two points a and b can be deformed over any point / without 
affecting the value of the function at b ; so that, when the preceding 
results are combined, the proposition enunciated is proved. 

By the continued application of the theorem, we are led to the following 
results : 

COROLLARY I. Whatever be the effect of the description of a circuit on the 
initial value of a function, a reversal of the circuit restores the original value 
of the function. 

For the circuit, when described positively and negatively, may be re 
garded as the contour of an area of infinitesimal breadth, which encloses no 
branch-point within itself and the description of the contour of which 
therefore restores the initial value of the function. 

COROLLARY II. A circuit can be deformed into any other circuit without 
affecting the final value of the function, provided that no branch-point be crossed 
in tJie process of deformation. 

It is thus justifiable, and it is often convenient, to deform a path con 
taining a single branch-point into a loop round the 
point. A loop* consists of a line nearly to the point, ~ 
nearly the whole of a very small circle round the point, Fig. 17. 

and a line back to the initial point; see figure 17. 

COROLLARY III. The value of a function is unchanged when the variable 
describes a closed circuit containing no branch-point ; it is likewise unchanged 
when the variable describes a closed circuit containing all the branch-points. 

The first part is at once proved by remarking that, without altering the 
value of the function, the circuit can be deformed into a point. 

For the second part, the simplest plan is to represent the variable on 
Neumann s sphere. The circuit is then a curve on the sphere enclosing all 
the branch-points : the effect on the value of the function is unaltered by any 
deformation of this curve which does make it cross a branch-point. The 
curve can, without crossing a branch-point, be deformed into a point in that 
other part of the area of the sphere which contains none of the branch 
points ; and the point, which is the limit of the curve, is not a branch 
point. At such a point, the value of the function is unaltered ; and there 
fore the description of a circuit, which encloses all the branch-points, 
restores the initial value of the function. 

COROLLARY IV. If the values of w at b for variations along two paths 

* French writers use the word lacet, German writers the word Schleife. 



154 



EFFECT OF DEFORMATION 



[90. 



acb, adb be not the same, then a description of acbda will not restore the initial 
value of w at a. 

In particular, let the path be the loop OeceO (fig. 17), and let it change w 
at into w . Since the values of w at are different and because there is 
no branch-point in Oe (or in the evanescent circuit OeO), the values of w at 
e cannot be the same : that is, the value with which the infinitesimal circle 
round a begins to be described is changed by the description of that circle. 
Hence the part of the loop that is effective for the change in the value of w is 
the small circle round the point ; and it is because the description of a small 
circle changes the value of w that the value of w is changed at after the 
description of a loop. 

If/0?) be the value of w which is changed mtof^z) by the description of 
the loop, so that/Oz) and f^(z) are the values at 0, then the foregoing 
explanation shews that /(e) and / (e) are the values at e, the branch /(e) 
being changed by the description of the circle into the branch /i(e). 

From this result the inference can be derived that the points B l} B. 2 , ... 
are branch-points as defined in 12. Let a be any one of the points, and 
let f(z) be the value of w which is changed into f, (z) by the description of 
a very small circle round a. Then as the branch of w is monogenic, the 
difference between f(z) and f^(z) is an infinitesimal quantity of the same 
order as the length of the circumference of the circle : so that, as the circle 
is infinitesimal and ultimately evanescent, \f(z) -/iOz)| can be made as small 
as we please with decrease of z - a or, in the limit, the values of /(a) and 
/(a) at the branch-point are equal. Hence each of the points B is such 
that two or more branches of the function have the same value at the point 
and there is interchange among these branches when the variable describes a 
small circuit round the point : which affords a definition of a branch-point, 
more complete than that given in 12. 

COROLLARY V. If a closed circuit contain several branch-points, the effect 
which it produces can be obtained by a combination of the effects produced in 
succession by a set of loops each going round only one of the branch-points. 

If the circuit contain several branch-points, say three as at a, b, c, then a 
path such as AEFD, in fig. 18, can without 
crossing any branch-point, be deformed into the 
loops AaB, BbC, GcD; and therefore the complete 
circuit AEFD A can be deformed validly into 
AaBbCcDA, and the same effect will be produced 
by the two forms of circuit. When D is made DA 

practically to coincide with A, the whole of the Fig. 18. 

second circuit is composed of the three loops. Hence the corollary. 

This corollary is of especial importance in the consideration of integrals 
of multiform functions. 




91.] OF PATH OF THE VAKIABLE 155 

COROLLARY VI. In a continuous part of the plane where there are no 
branch-points, each branch of a multiform function is uniform. 

Each branch is monogenic and, except at isolated points, continuous; 
hence, in such regions of the plane, all the propositions which have been 
proved for monogenic analytic functions can be applied to each of the 
branches of a multiform function. 

91. If there be a branch-point within the circuit, then the value of the 
function at 6 consequent on variations along acb may, but will not necessarily, 
differ from its value at the same point consequent on variations along adb. 
Should the values be different, then the description of the whole curve acbda 
will lead at a not to the initial value of w, but to a different value. 
The test as to whether such a change is effected by the description is 
immediately derivable from the foregoing proposition; and as in Corollary 
IV., 90, it is proved that the value is or is not changed by the loop, 
according as the value of w for a point near the circle of the loop is 
or is not changed by the description of that circle. Hence it follows that, if 
there be a branch-point which affects the branch of the function, a path of 
variation of the independent variable cannot be deformed across the branch 
point without a change in the value of w at the extremity of the path. 

And it is evident that a point can be regarded as a branch-point for a 
function only if a circuit round the point interchange some (or all) of the 
branches of the function which are equal at the point. It is not necessary that 
all the branches of the function should be thus affected by the point : it is 
sufficient that some should be interchanged*. 

Further, the change in the value of w for a single description of a circuit 
enclosing a branch-point is unique. 

For, if a circuit could change w into w or w", then, beginning with w" 
and describing it in the negative sense we should return to w and afterwards 
describing it in the positive sense with w as the initial value we should 
obtain w . Hence the circuit, described and then reversed, does not restore 
the original value w" but gives a different branch w ; and no point on 
the circuit is a branch-point. This result is in opposition to Corollary I., 
of 90 ; and therefore the hypothesis of alternative values at the end of 
the circuit is not valid, that is, the change for a single description is 
unique. 

But repetitions of the circuit may, of course, give different values at the 
end of successive descriptions. 

* In what precedes, certain points were considered which were regular singularities (see 
p. 163, note) and certain which were branch-points. Frequently points will occur which are at 
once branch-points and infinities ; proper account must of course be taken of them. 



156 



LAW OF INTERCHANGE 



[92. 




Fig. 19. 



92. Let be any ordinary point of the function ; join it to all the 
branch-points (generally assumed finite in 
number) in succession by lines which do not 
meet each other : then each branch is uniform 
for each path of variation of the variable which 
meets none of these lines. The effects pro 
duced by the various branch-points and their 
relations on the various branches can be indi 
cated by describing curves, each of which 
begins at a point indefinitely near and 
returns to another point indefinitely near it 
after passing round one of the branch- points, 
and by noting the value of each branch of the function after each of these 
curves has been described. 

The law of interchange of branches of a function after description of a 
circuit round a branch-point is as follows: 

All the branches of a function, which are affected by a branch-point as such, 
can either be arranged so that the order of interchange (for description of a 
path round the point) is cyclical, or be divided into sets in each of which the 
order of interchange is cyclical. 

Let w lt w. 2> w 3) ... be the branches of a function for values of z near a 
branch-point a which are affected by the description of a small closed curve 
C round a : they are not necessarily all the branches of the function, but only 
those affected by the branch-point. 

The branch w^ is changed after a description of C ; let w 2 be the branch 
into which it is changed. Then w 2 cannot be unchanged by C; for a reversed 
description of C, which ought to restore w 1} would otherwise leave w. 2 un 
changed. Hence w 2 is changed after a description of (7; it may be changed 
either into w 1 or into a new branch, say w 3 . If into w lt then w-^ and w 2 form 
a cyclical set. 

If the change be into w 3 , then w 3 cannot remain unchanged after a 
description of C, for reasons similar to those that before applied to the 
change of w. 2 : and it cannot be changed into w 2 , for then a reversed de 
scription of G would change w z into w. A , and it ought to change w 2 into w^ 
Hence, after a description of C, w 3 is changed either into w^ or into a new 
branch, say w 4 . If into w 1} then w 1} w 2 , w 3 form a cyclical set. 

If the change be into w 4 , then w 4 cannot remain unchanged after a 
description of G ; and it cannot be changed into w. 2 or w s , for by a reversal 
of the circuit that earlier branch would be changed into w 4 whereas it ought 
to be changed into the branch, which gave rise to it by the forward descrip 
tion a branch which is not w 4 . Hence, after a description of C, w 4 is 
changed either into w^ or into a new branch. If into w lf then w j} w. 2 , w 3 , w 4 
form a cyclical set. 



92.] OF BRANCHES OF A FUNCTION 157 

If w 4 be changed into a new branch, we proceed as before with that new 
branch and either complete a cyclical set or add one more to the set. By 
repetition of the process, we complete a cyclical set sooner or later. 

If all the branches be included, then evidently their complete system 
taken in the order in which they come in the foregoing investigation is a 
system in which the interchange is cyclical. 

If only some of the branches be included, the remark applies to the set 
constituted by them. We then begin with one of the branches not included 
in that set and evidently not inclusible in it, and proceed as at first, until 
we complete another set which may include all the remaining branches or 
only some of them. In the latter case, we begin again with a new branch 
and repeat the process ; and so on, until ultimately all the branches are 
included. The whole system is then arranged in sets, in each of which the 
order of interchange is cyclical. 

93. The analytical test of a branch-point is easily obtained by con 
structing the general expression for the branches of a function which are 
interchanged there. 

Let z = a be a branch-point where n branches w 1} ^v 2 ,..., w n are cyclically 
interchanged. Since by a first description of a small curve round a, the 
branch w 1 changes into w 2 , the branch w into w s , and so on, it follows that 
by r descriptions w 1 is changed into w r+l and by n descriptions w l reverts to 
its initial value. Similarly for each of the branches. Hence each branch 
returns to its initial value after n descriptions of a circuit round a branch 
point where n branches of the function are interchangeable. 

Now let z - a = Z n ; 

then, when z describes circles round a, Z moves in a circular arc round its 
origin. For each circumference described by z, the variable Z describes 

-th part of its circumference; and the complete circle is described by Z 
round its origin when n complete circles are described by z round a. Now 
the substitution changes w r as a function of z into a function of Z, say into 
W r ; and, after n complete descriptions of the ^-circle round a, w r returns 
to its initial value. Hence, after the description of a ^-circle round its 
origin, W r returns to its initial value, that is, Z = ceases to be a branch 
point for W r . Similarly for all the branches W. 

But no other condition has been associated with a as a point for the 
function w ; and therefore Z = may be any point for the function W, that 
is, it may be an ordinary point, or a singularity. In every case we have W 
a uniform function of Z in the immediate vicinity of the origin ; and therefore 
in that vicinity it can be expressed in the form 



158 ANALYTICAL EXPRESSION [93. 

with the significations of P and G already adopted. When Z is an ordinary 
point, G is a constant or zero ; when Z is an accidental singularity, O is an 
algebraical function ; and, when Z is an essential singularity, G is a transcen 
dental function. 

The simpler cases are, of course, those in which the form of G is alge 
braical or constant or zero ; and then W can be put into the form 

Z m P(Z), 

where P is an infinite series of positive powers and m is an integer. As this 
is the form of W in the vicinity of Z=Q, it follows that the form of w in the 
vicinity of z = a is 

m 1 

(z - a) n P {(z - a) n } 

and the various n branches of the function are easily seen to be given by 

i 
substituting in the above for (z a) n the values 

2im j. 

e m (z of, 

where s = 0, 1,..., n 1. We therefore infer that the general expression for 
the n branches of a function, which are interchanged by circuits round a 
branch-point z = a, assumed not to be an essential singularity, is 

m _ 1 

(z - a) Tl P {(z - a)}, 

i 

where m is an integer, and where to (z a) n its n values are in turn assigned 
to obtain the different branches of the function. 

There may be, however, more than one cyclical set of branches. If there 
be another set of r branches, then it may similarly be proved that their 

general expression is 

OTI _ i 

(zjaYQ{(z-ay-}, 

where m^ is an integer, and Q is an integral function ; the various branches 

i 
are obtained by assigning to (z a) r its r values in turn. 

And so on, for each of the sets, the members of which are cyclically 
interchangeable at the branch-point. 

When the branch-point is at infinity, a different form is obtained. Thus 
in the case of a set of n cyclically interchangeable branches we take 

z = %-, 

so that n negative descriptions of a closed -curve, excluding infinity and no 
other branch-point, requires a single positive description of a closed curve 
round the w-origin. These n descriptions restore the value of w; as a function 
of z to its initial value; and therefore the single description of the M- curve 
round the origin restores the value of U the equivalent of w after the 



93.] NEAR A BRANCH-POINT 159 

change of the independent variable as a function of u. Thus u = ceases 
to be a branch-point for the function U ; and therefore the form of U is 



. . 

where the symbols have the same general signification as before. 

If, in particular, z = oo be a branch-point but not an essential singularity, 
then G is either a constant or an algebraical function ; and then U can be 
expressed in the form 

u~ m P(u}, 

where TO is an integer. When the variable is changed from u to z, then the 
general expression for the n branches of a function which are interchangeable 

at z = oo , assumed not to be an essential singularity, is 



where TO is an integer and where to z n its n values are assigned to obtain the 
different branches of the function. 

If, however, the branch-point z = a in the former case or z = oo in the 
latter be an essential singularity, the forms of the expressions in the vicinity 
of the point are 

_i i 

G{(z-a) J>-p{(jr-aJ} f 

i _i 

and G(z n ) + P(z n ), 

respectively. 

Note. When a multiform function is denned, either explicitly or im 
plicitly, it is practically always necessary to consider the relations of the 
branches of the function for z = oo as well as their relations for points that 
are infinities of the function. The former can be determined by either 
of the processes suggested in 4 for dealing with z=<x>; the latter can be 
determined as in the present article. 

Moreover, the total number of branches of the function has been assumed 
to be finite. The cases, in which the number of branches is unlimited, need 
not be discussed in general : it will be sufficient to consider them when they 
arise, as they do arise, e.g., when the function is of the form of an algebraical 
irrational with an irrational index such as z^ hardly a function in the 
ordinary sense, or when the function is the logarithm of a function of z, 
or is the inverse of a periodic function. In the nature of their multiplicity 
of branching and of their sequence of interchange, they are for the most part 
distinct from the multiform functions with only a finite number of branches. 

Ex. The simplest illustrations of multiform functions are furnished by functions 
denned by algebraical equations, in particular, by algebraic irrationals. 



160 



ALGEBRAICAL 



[93. 



The general type of the algebraical irrational is the product of a number of functions 
of the form w = {A(z a l )(z-a. 2 ) ...... (z-a^)} m , m and n being integers. 

This particular function has m branches; the points a 1} 2 , ...... , a n are branch-points. 

To find the law of interchange, we take z-a r = pe 01 ; then when a small circle of radius p 
is described round a r , so that z returns to its initial position, the value of 6 increases by 

2n and the new value of w is aw, where a is the with root of unity defined by e m m . Taking 
then the various branches as given by w, aw, a?w, ...... , a m ~ l w, we have the law of inter 

change for description of a small curve round any one-branch point as given by this 
succession in cyclical order. The law of succession for a circuit enclosing more than 
one of the branch-points is derivable by means of Corollary V, 90. 

To find the relation of z = o> to w, we take zz = l and consider the new function W in 
the vicinity of the ^ -origin. We have 

W ={A (1 -VH1 -a/) ...... (l-a n <)}^ ~. 

If the variable z 1 describe a very small circle round the origin in the negative sense, then 

27TZ 

z is multiplied by e~ 2 and so W acquires a factor e , that is, W is changed unless 
this acquired factor is unity. It can be unity only when n/m is an integer ; and therefore, 
except when n/m is an integer, 0=00 is a branch-point of the function. The law of 
succession is the same as that for negative description of the z -circle, viz., w, a n w, 
a 2n w, ...... ; the m values form a single cycle only if n be prime to m, and a set of cycles 

if n be not prime to m. 

Thus 0=00 is a branch -point for w = (k?-gg-g^~^ ; it is not a branch-point for 
w = {(\ -2 2 ) (1 & 2 z 2 )}~*; and z = b is a branch-point for the function defined by 

(z b) w 2 = z a, 
but z = b is riot a branch-point for the function defined by (zb) 2 w z = z-a. 

Again, if p denote a particular value of f t when z has a given value, and q similarly 

denote a particular value of [ : ) , then w=p+q is a six-valued function, the values 

V+v 
being 



W 6 = -p + aq, 

where a is a primitive cube root of unity. The branch-points are - 1, 0, 1, oo ; and the 
orders of change for small circuits round one (and only one) of these points are as 
follows : 



For a small circuit round 


-1 





1 


00 


Wj changes to 


ft 


W-i 


W 3 


W 2 


W 2 


w e 


W 1 


W 4 


w, 


^3 


to, 


W 4 


W 5 


W 4 


W 4 


M> 2 


w, 


W 6 


W 3 


Ws n 


W 3 


W 6 


w, 


W 6 


U>6 


W7 4 


W 5 


Wo 


W 5 



93.] FUNCTIONS 161 

Combinations can at once be effected ; thus, for a positive circuit enclosing both 1 and QO 
but* not 1 or 0, the succession is 

io lt w 4 , w 6 , w 2 , w 3 , W G 
in cyclical order. 

94. It has already been remarked that algebraic irrationals are a special 
class of functions denned by algebraical equations. Functions thus generally 
denned by equations, which are algebraical so far as concerns the dependent 
variable but need not be so in reference to the independent variable, are 
often called algebraical. The term, in one sense, cannot be strictly applied 
to the roots of an equation of every degree, seeing that the solution 
of equations of the fifth and higher degrees can be effected only by 
transcendental functions; but what is implied is that a finite number of 
determinations of the dependent variable is given by the equation -f*. 

The equation is algebraical in relation to the dependent variable w, that 
is, it will be taken to be of finite degree n in w. The coefficients of the 
different powers will be supposed to be rational uniform functions of z : were 
they irrational in any given equation, the equation could be transformed 
into another, the coefficients of which are rational uniform functions. And 
the equation is supposed to be irreducible, that is, if the equation be taken 
in the form 

f(w, *) = 0, 

the left-hand member f(w, z) cannot be resolved into factors of a form and 
character as regards w and z similar to /itself. 

The existence of equal roots of the equation for general values of z 
requires that 

fi \ j "df( w > z ) 
f(w,z) and ^~ 

shall have a common factor, which will be rational owing to the form of 
f(w, z}. This form of factor is excluded by the irreducibility of the equation ; 
so that /= 0, as an equation in w, has not equal roots for general values 
of z. But though the two equations are not both satisfied in virtue of a 
simpler equation, they are two equations determining values of w and #; 
and their form is such that they will give equal values of w for special 
values of z. 

Since the equation is of degree n, it may be taken to be 



w 



where the functions F 1} F 2} ... are rational and uniform. If all their singu- 

* Such a circuit, if drawn on the Neumann s sphere, may be regarded as excluding - 1 and 0, 
or taking account of the other portion of the surface of the sphere, it may be regarded as a 
negative circuit including - 1 and 0, the cyclical interchange for which is easily proved to be 
iCj, w 4 , w 5 , w. 2 , M? 3 , w 6 as in the text. 

t Such a function is called Men defini by Liouville. 

F. 11 



162 ALGEBRAICAL [94. 

larities be accidental, they are raeromorphic algebraical functions of z (unless 
z = oo is the only singularity, in which case they are holomorphic) ; and the 
equation can then be replaced by one which is equivalent and has all its 
coefficients holomorphic, the coefficient of w n being the least common multiple 
of all the denominators of the meromorphic functions in the first form. This 
form cannot however be deduced, if any of the singularities be essential. 

The equation, as an equation in w, has n roots, all functions of z ; let 
these be denoted by w 1 ,w 2 ,..., iv n , which are the n branches of the function w. 
When the geometrical interpretation is associated with the analytical relation, 
there are n points in the w-plane, say a 1 ,..., a n , which correspond with a point 
in the ^-plane, say with c^ ; and in general these n points are distinct. As 
z varies so as to move in its own plane from a, then each of the w-points 
moves in their common plane ; and thus there are n w-paths corresponding 
to a given z-path. These n curves may or may not meet one another. 

If they do not, there are n distinct w-paths, leading from a 1; ..., a n to 
/3i,..., /3 n , respectively corresponding to the single ^-path leading from a 
to b. 

If two or more of the w-paths do meet one another, and if the describing 
w-poirits coincide at their point of intersection, then at such a point of 
intersection in the w-plane, the associated branches w are equal ; and 
therefore the point in the ^-plane is a point that gives equal values for w. 
It is one of the roots of the equation obtained by the elimination of w 
between 



the analytical test as to whether the point is a branch-point will be 
considered later. The march of the concurrent ^-branches from such a 
point of intersection of two w-paths depends upon their relations in its 
immediate vicinity. 

When no such point lies on a ^-path from a to b, no two of the w-points 
coincide during the description of their paths. By 90, the 2-path can be 
deformed (provided that, in the deformation, it does not cross a branch-point) 
without causing any two of the w-points to coincide. Further, if z describe 
a closed curve which includes none of the branch -points, then each of the 
^-branches describes a closed curve and no two of the tracing points ever 
coincide. 

Note. The limitation for a branch-point, that the tracing w-points 
coincide at the point of intersection of the w-curves, is of essential im 
portance. 

What is required to establish a point in the z-plane as a branch-point, 
is not a mere geometrical intersection of a couple of completed w-paths but 
the coincidence of the w-points as those paths are traced, together with inter- 



94.] FUNCTIONS 1 63 

change of the branches for a small circuit round the point. Thus let there be 
such a geometrical intersection of two w-curves, without coincidence of the 
tracing points. There are two points in the ^-plane corresponding to the 
geometrical intersection ; one belongs to the intersection as a point of the 
w-paih which first passed through it, and the other to the intersection as a 
point of the w-path which was the second to pass through it. The two 
branches of w for the respective values of z are undoubtedly equal ; but the 
equality would not be for the same value of z. And unless the equality 
of branches subsists for the same value of z, the point is not a branch 
point. 

A simple example will serve to illustrate these remarks. Let w be defined by the 
equation 



so that the branches w 1 and w 2 are given by 

Ci0j_ = cz+z(z 2 + c 2 )*, cw 2 = cz-z(z*-\- c 2 )* ; 
it is easy to prove that the equation resulting from the elimination of w between /=0 and 



and that only the two points z= ic are branch-points. 

The values of z which make w l equal to the value of w z for z = a (supposed not equal to 
either 0, ci or ci) are given by 

cz + z (0 2 + c 2 )* = ca - a (a 2 + c 2 )*, 

which evidently has not 2 = a for a root. Rationalising the equation so far as concerns z 
and removing the factor z -a, as it has just been seen not to furnish a root, we find that s 
is determined by 

z 3 + z 2 a + za 2 + a 3 + 2ac 2 - 2ac (a 2 + c 2 ) * = 0, 

the three roots of which are distinct from a, the assumed point, and from ci, the branch 
point. Each of these three values of z will make w v equal to the value of w 2 for z=a : we 
have geometrical intersection without coincidence of the tracing points. 

95. When the characteristics of a function are required, the most im 
portant class are its infinities: these must therefore now be investigated. 
It is preferable to obtain the infinities of the function rather than the 
singularities alone, in the vicinity of which each branch of the function 
is uniform * : for the former will include these singularities as well as 
those branch-points which, giving infinite values, lead to regular singularities 
when the variables are transformed as in 93. The theorem which deter 
mines them is: 

The infinities of a function determined by an algebraical equation are the 
singularities of the coefficients of the equation. 
Let the equation be 

w n + w n-i FI ^ + w n-, !,(*) + ... + rf^ (^) + ^ (^) = Q, 

* These singularities will, for the sake of brevity, be called regular. 

112 



164 INFINITIES [95. 

and let w be any branch of the function; then, if the equation which 
determines the remaining branches be 

w n-i + w n- 2 Q i ^ + w n-3 2 (Y) + . . . + W G n -z ( Z ) + G n -i (z) = 0, 

we have F n (z) = - w G n -i (z), 

F n ^ (z) = - w Gn-z (z) + _! (z), 

^71-2 (Z) = - w Gn-s (z} + #n-2 (z), 



Now suppose that a is an infinity of w ; then, unless it be a zero of order 
at least equal to that of G n ^ (z), a is an infinity of F n (z). If, however, it be 
a zero of G n -i (z) of sufficient order, then from the second equation it is an 
infinity of F n _ l (z) unless it is a zero of order at least equal to that of 
6r n _ 2 (z) ; and so on. The infinity must be an infinity of some coefficient not 
earlier than Fi (z) in the equation, or it must be a zero of all the functions 
G which are later than Gf_! (z). If it be a zero of all the functions G r , so 
that we may not, without knowing the order, assert that it is of rank at 
least equal to its order as an infinity of w , still from the last equation it 
follows that a must be an infinity of F l (z). Hence any infinity of w is an 
infinity of at least one of the coefficients of the equation. 

Conversely, from the same equations it follows that a singularity of one 
of the coefficients is an infinity either of w or of at least one of the co 
efficients G. Similarly the last alternative leads to an inference that the 
infinity is either an infinity of another branch w" or of the coefficients of the 
(theoretical) equation which survives when the two branches have been 
removed. Proceeding in this way, we ultimately find that the infinity either 
is an infinity of one of the branches or is an infinity of the coefficient in the 
last equation, that is, of the last of the branches. Hence any singularity 
of a coefficient is an infinity of at least one of the branches of the function. 

It thus appears that all the infinities of the function are included among, 
and include, all the singularities of the coefficients ; but the order of the 
infinity for a branch does not necessarily make that point a regular 
singularity nor, if it be made a regular singularity, is the order necessarily 
the same as for the coefficient. 

96. The following method is effective for the determination of the order 
of the infinity of the branch. 

Let a be an accidental singularity of one or more of the F functions, 
say of order ra; for the function F t ; and assume that, in the vicinity of a, 
we have 

F t (z) = (z- a)-* [ Ci + di (z-a) + e { (z - a? +...]. 



96.] 



OF ALGEBRAICAL FUNCTIONS 



165 




Then the equation which determines the first term of the expansion of w in 
a series in the vicinity of a is 

w n + d (z a)~i w n ~ l + c 2 (z a)~ m 2 w n ~ 2 + ... 

-f c n _! (z - a)~ m -i w + c n (z - a)~ m = 0. 

Mark in a plane, referred to two rectangular axes, points n, 0; n 1, 
m^; n 2, m 2 ; . . ., 0, m n ; let these 
be A , A 1} ..., A n respectively. Any line 
through Ai has its equation of the form 

1 1 I nm ~~ ~\ J o" l w T. t)\\ 
y T *< A, {J, (71, tftf 

that is, 

y \x = \ (n i) mi. 

If then w = (z a)~ x f(z}, where f(z) is 
finite when z = a, the intercept of the fore 
going line on the negative side of the axis of y is equal to the order of the 
infinity in the term 

w n - i F i (z). 

This being so, we take a line through A n coinciding in direction with the 
negative part of the axis of y and we turn it about A n in a trigonometrically 
positive direction until it first meets one of the other points, say A n _ r ; then 
we turn it about A n _ r until it meets one of the other points, say A n _ s ; and 
so on until it passes through A . There will thus be a line from A n to 
A , generally consisting of a number of parts ; and none of the points A 
will be outside it. 

The perpendicular from the origin on the line through A n _ r and A n _ g is 
evidently greater than the perpendicular on any parallel line through a 
point A, that is, on any line through a point A with the same value 
of X; and, as this perpendicular is 

it follows the order of the infinite terms in the equation, when the particular 
substitution is made for w, is greater for terms corresponding to points lying 
on the line than it is for any other terms. 

If /(*) = wnen z = a, then the terms of lowest order after the substitu 
tion of (z a)~ K f(z) for w are 

as many terms occurring in the bracket as there are points A on the line 
joining A n _r to A n _ s . Since the equation determining w must be satisfied, 
terms of all orders must disappear, and therefore 



an equation determining s-r values of 6, that is, the first terms in the 
expansions of s r branches w. 



166 INFINITIES [96. 

Similarly for each part of the line : for the first part, there are r branches 
with an associated value of X ; for the second, s r branches with another 
associated value ; for the third, t s branches with a third associated value ; 
and so on. 

The order of the infinity for the branches is measured by the tangent 
of the angle which the corresponding part of the broken line makes with the 
axis of a; ; thus for the line joining A n ^. to A n _ s the order of the infinity for 
the s - r branches is 



where m n _ r and m n _ s are the orders of the accidental singularities of F n _ r (z) 
and F n _ s (z). 

If any part of the broken line should have its inclination to the axis of 
x greater than \ir so that the tangent is negative and equal to - //,, then the 
form of the corresponding set of branches w is (z a,y g {z} for all of them, 
that is, the point is not an infinity for those branches. But when the 
inclination of a part of the line to the axis is < \TT, so that the tangent is 
positive and equal to X, then the form of the corresponding set of branches 
w is (z a)~ K f(z) for all of them, that is, the point is an infinity of order X 
for those branches. 

In passing from A n to A there may be parts of the broken line which 
have the tangential coordinate negative, implying therefore that a is not an 
infinity of the corresponding set or sets of branches w. But as the revolving 
line has to change its direction from A n y to some direction through A , 
there must evidently be some part or parts of the broken line which have 
their tangential coordinate positive, implying therefore that a is an infinity 
of the corresponding set or sets of branches. 

Moreover, the point a is, by hypothesis, an accidental singularity of at 
least one of the coefficients and it has been supposed to be an essential 
singularity of none of them; hence the points A , A 1} ..., A n are all in the 
finite part of the plane. And as no two of their abscissa are equal, no line 
joining two of them can be parallel to the axis of y, that is, the inclination 
of the broken line is never \ir and therefore the tangential coordinate is 
finite, that is, the order of the infinity for the branches is finite for any 
accidental singularity of the coefficients. 

If the singularity at a be essential for some of the coefficients, the 
corresponding result can be inferred by passing to the limit which is 
obtained by making the corresponding value or values of m infinite. In 
that case the corresponding points A move to infinity and then parts of the 
broken line pass through A (which is always on the axis of x) parallel to 
the axis of y, that is, the tangential coordinate is infinite and the order of 



96.] OF ALGEBRAICAL FUNCTIONS 167 

the infinity at a for the corresponding branches is also infinite. The point is 
then an essential singularity (and it may be also a branch-point). 

It has been assumed implicitly that the singularity is at a finite point in 
the 2-plane ; if, however, it be at oo , we can, by using the transformation 
zz 1 and discussing as above the function in vicinity of the origin, obtain 
the relation of the singularity to the various branches. We thus have the 
further proposition : 

The order of ike infinity of a branch of an algebraical function at a 
singularity of a coefficient of the equation, which determines the function, is 
finite or infinite according as the singularity is accidental or essential. 

If the coefficients FI of the equation be holomorphic functions, then 
z = oo is their only singularity and it is consequently the only infinity for 
branches of the function. If some of or all the coefficients F f be mero- 
morphic functions, the singularities of the coefficients are the zeros of 
the denominators and, possibly, =oo; and, if the functions be algebraical, 
all such singularities are accidental. In that case, the equation can be 
modified to 

h (z) w n + h^ (z} w n ~ l + A 2 (z) w n ~ 2 + . . . = 0, 

where h (z) is the least common multiple of all the denominators of the 
functions F t . The preceding results therefore lead to the more limited 
theorem : 

When a function w is determined by an algebraical equation the coefficients 
of which are holomorphic functions of z, then each of the zeros of the coefficient 
of the highest power of w is an infinity of some of (and it may be of all) the 
branches of the function w, each such infinity being of finite order. The point 
z= oo may also be an infinity of the function w ; the order of that infinity is 
finite or infinite according as z = oo is an accidental or an essential singularity 
of any of the coefficients. 

It will be noticed that no precise determination of the forms of the 
branches w at an infinity has been made. The determination has, however, 
only been deferred : the infinities of the branches for a singularity of the 
coefficients are usually associated with a branch-point of the function and 
therefore the relations of the branches at such a point will be of a general 
character independent of the fact that the point is an infinity. 

If, however, in any case a singularity of a coefficient should prove to be, 
not a branch-point of w but only a regular singularity, then in the vicinity of 
that point the branch of w is a uniform function. A necessary (but not suffi 
cient) condition for uniformity is that (m n _r m n _ s ) -7- (s r) be an integer. 

Note. The preceding method can be applied to determine the leading 
terms of the branches in the vicinity of a point a which is an ordinary point 
for each of the coefficients F. 



168 BRANCH-POINTS [97. 

97. There remains therefore the consideration of the branch-points of a 
function determined by an algebraical equation. 

The characteristic property of a branch-point is the equality of branches 
of the function for the associated value of the variable, coupled with the 
interchange of some of (or all) the equal branches after description by the 
variable of a small contour enclosing the point. 

So far as concerns the first part, the general indication of the form of the 
values has already ( 93) been given. The points, for which values of w 
determined as a function of z by the equation 

f(w, z) =J 

are equal, are determined by the solution of this equation treated simul 
taneously with 

df(w, z) = Q . 
dw 

and when a point z is thus determined the corresponding values of w, which 
are equal there, are obtained by substituting that value of z and taking M, 

the greatest common measure of / and -J- . The factors of M then lead to 

the value or the values of w at the point ; the index m of a linear factor 
gives at the point the multiplicity of the value which it determines, and 
shews that m + 1 values of w have a common value there, though they are 
distinct at infinitesimal distances from the point. If m = 1 for any factor, 
the corresponding value of w is an isolated value and determines a branch 
that is uniform at the point. 

Let z = a, w = a be a value of z and a value of w thus obtained ; and 
suppose that m is the number of values of w that are equal to one another. 
The point z = a is not a branch-point unless some interchange among the 
in values of w is effected by a small circuit round a ; and it is therefore 
necessary to investigate the values of the branches* in the vicinity of z a. 

Let w = a. + w , z = a + z ; then we have 



that is, on the supposition that f(w, z) has been freed from fractions, 

/(a, a) + SS^ r XV = 0, 

r, s 

so that, since a is a value of w corresponding to the value a of z, we have 
w and / connected by the relation 



* The following investigations are founded on the researches of Puiseux on algebraic 
functions; they are contained in two memoirs, Liouville, l re Ser., t. xv, (1850), pp. 365 480, ib., 
t. xvi, (1851), pp. 228240. See also the chapters on algebraic functions, pp. 19 76, in the 
second edition of Briot and Bouquet s Theorie des fonctions elliptiques. 



97.] OF ALGEBRAICAL FUNCTIONS 169 

When / is 0, the zero value of w must occur m times, since a is a root 
m times repeated; hence there are terms in the foregoing equation inde 

pendent of z, and the term of lowest index among them is w m . Also when 
w = 0, z is a possible root ; hence there must be a term or terms 
independent of w in the equation. 

First, suppose that the lowest power of z among the terms independent 
of w is the first. The equation has the form 

Az + higher powers of z 
+ Biu + higher powers of w 
+ terms involving z and w = 0, 

O-/* / \ 

where A is the value of - - for w = a, z = a. Let z =% m , w = v%: the 

02 

last form changes to 

(A + Bv m ) m + terms with m+1 as a factor = ; 
and therefore A + Bv m + terms involving = 0. 

Hence in the immediate vicinity of z = a, that is, of =0, we have 

A + Bv = 0. 

Neither A nor B is zero, so that all the m values of v are finite. Let them 
be v l} ..., v m , so arranged that their arguments increase by 2-Tr/Tn through 
the succession. The corresponding values of w are 



for i = l, ..., m. Now a ^-circuit round a, that is, a /-circuit round its 
origin, increases the argument of z by 2?r ; hence after such a circuit we 

1_ 27Tt !_ 

have the new value of w{ as ViZ /m e m , that is, it is v i+1 z m which is the value 
of w i+l . Hence the set of values w\, /.,,..., w m form a complete set of 
interchangeable values in their cyclical succession ; all the m values, which 
are equal at a, form a single cycle and the point is a branch-point. 

Next, suppose that the lowest power of z among the terms independent 
of w is z , where I > 1. The equation now has the form 
= Az + higher powers of z 
+ Bw + higher powers of w 






A rs z V 



r=l s=l 

where in the last summation r and s are not zero and in every term either 
(i), r is equal to or greater than I or (ii), s is equal to or greater than m 
or (iii), both (i) and (ii) are satisfied. As only terms of the lowest orders 



170 



BRANCH-POINTS 



[97. 



need be retained for the present purpose, which is the derivation of the first 
term of w in its expansion in powers of z , we may use the foregoing equation 
in the form 



, l-lm-l 

A/ + 2 2 A, 

r=l s=l 



,r ,s , -p. ,m _ 

jf w + Bw = 0. 



To obtain this first term we proceed in a manner similar to that in 96 *. 
Points A ,..., A m are taken in a plane 
referred to rectangular axes having as co 
ordinates 0, ;...; s, r;...; m, respectively. 
A line is taken through A m and is made to 
turn round A m from the position A m O until 
it first meets one of the other points ; then 
round the last point which lies in this 
direction, say round Aj, until it first meets 
another ; and so on. 

Any line through A t (the point s i} r t ) is 
of the form 

y - Ti = - \ (x - s^. 

The intercept on the axis of /-indices is \Si + Ti, that is, the order of the 
term involving A rs for a substitution w oc / . The perpendicular from the 
origin for a line through AI and Aj is less than for any parallel line through 
other points with the same inclination ; and, as this perpendicular is 




Fig. 21. 



it follows that, for the particular substitution w oc z , the terms corresponding 
to the points lying on the line with coordinate X are the terms of lowest 
order and consequently they are the terms which give the initial terms for 
the associated set of quantities w . 

Evidently, from the indices retained in the equation, the quantities X 
for the various pieces of the broken line from A m to A are positive and 
finite. 

Consider the first piece, from A m to Aj say ; then taking the value of X for 

that piece as fa, so that we write v^z * 1 as the first term of w , we have as the 
set of terms involving the lowest indices 

J? /"* i ^ ^ A fl* fi I A fl*i , ^J 

Sj being the smallest value of s retained ; and then 



so that 



/*! = 



m s 



* Reference in this connection may be made to Chrystal s Algebra, ch. xxx., with great 
advantage, as well as the authorities quoted on p. 168, note. 



GROUPING OF BRANCHES 171 

Let p/q be the equivalent value of ^ as the fraction in its lowest terms ; and 

p 

write / = (?. Then w = v l z i = vtf ; all the terms except the above group 

are of order > mp and therefore the equation leads after division by % m Ptfi to 

Bv^- i + ^A ra vf-*i + A rfj = 0, 

an equation which determines m Sj values for v l , and therefore the initial 
terms of m Sj of the w-branches. 

Consider now the second piece, from Aj to At say ; then taking the value 

of A, for that piece as fa, so that we write v. 2 z ^ as the first term of w , we 
have as the set of terms involving the lowest indices for this value of /* 2 

A f r i / s .- xr"O A i v I s fl"i t s i 

A rf z J w + E&A.rjt w + A r . s z *w \ 
where S{ is the smallest value of s retained. Then 

Sjfr + T j = tyig + r 

Proceeding exactly as before, we find 



as the equation determining Sj-Si values for v 2 and therefore the initial 
terms of Sj s t of the w-branches. 

And so on, until all the pieces of the line are used ; the initial terms of 
all the w-branches are thus far determined in groups connected with the 
various pieces of the line A^Ai^.A,. By means of these initial terms, 
the m-branches can be arranged for their interchanges, by the description of 
a small circuit round the branch-point, according to the following theorem : 

Each group can be resolved into systems, the members of each of which are 
cyclically interchangeable. 

It will be sufficient to prove this theorem for a single group, say the 
group determined by the first piece of broken line: the argument is 
general. 

Since - is the equivalent of ^ and of T} . and since s, < s, we have 
V m s m Sj 

m-s = kq, m-Sj^kjq, kj>k; 
and then the equation which determines ^ is 

Sv&v + 2^ r ,, Vl <*,-*> 1 4 Ar jSj = 0, 

that is, an equation of degree k } in vj as its variable. Let U be any root of 
it ; then the corresponding values of v l are the values of U*. Suppose these 
q values to be arranged so that the arguments increase by 27r^, which is 

possible, because p is prime to q. Then the q values of w being the values 
of v^Vi are 

P. P P 



172 GROUPING OF BRANCHES [97. 

where v la is that value of Ifi which has for its argument. A circuit 

round the /-origin evidently increases the argument of any one of these 
w -values by Zrrp/q, that is, it changes it into the value next in the succession; 
and so the set of q values is a system the members of which are cyclically 
interchangeable. 

This holds for each value of U derived from the above equation ; so that 
the whole set of m Sj branches are resolved into kj systems, each containing 
q members with the assigned properties. 

It is assumed that the above equation of order kj in vj has its roots unequal. 
If, however, it should have equal roots, it must be discussed ab initio by a 
method similar to that for the general equation; as the order kj (being a 
factor of m Sj) is less than m, the discussion will be shorter and simpler, 
and will ultimately depend on equations with unequal roots as in the case 
above supposed. 

It may happen that some of the quantities /j, are integers, so that the 
corresponding integers q are unity : a number of the branches would then be 
uniform at the point. 

It thus appears that z = a is a branch-point and that, under the present 
circumstances, the m branches of the function can be arranged in systems, 
the members of each one of which are cyclically interchangeable. 

Lastly, it has been tacitly assumed in what precedes that the common 
value of w for the branch-point is finite. If it be infinite, this infinite value 
can, by 95, arise only out of singularities of the coefficients of the equation : 
and there is therefore a reversion to the discussion of 95, 96. The dis 
tribution of the various branches into cyclical systems can be carried out 
exactly as above. 

Another method of proceeding for these infinities would be to take 
ww = \, z= c + z ; but this method has no substantial advantage over the 
earlier one and, indeed, it is easy to see that there is no substantial 
difference between them. 

Ex. 1. As an example, consider the function determined by the equation 



The equation determining the values of z which give equal roots for w is 

82 (2 -1) 2 = 4(3 -I) 3 
so that the values are z=l (repeated) and z= 1. 

When z=l, then w=0, occurring thrice; and, if 2 = 1+2 then 

8W/3W, 

that is, w ^^z 13 . 

The three values are branches of one system in cyclical order for a circuit round z=\. 



97.] EXAMPLES 173 

When z = 1, the equation for w is 



that is, (w 

so that w=\ or w= , occurring twice. 

For the former of these we easily find that, for s= l-\-z , the value of w is 
l-hfs -f ...... , an isolated branch as is to be expected, for the value 1 is not 

repeated. 

For the latter we take w \ + w and find 

so that the two branches are 



and they are cyclically interchangeable for a small circuit round z= - 1. 

These are the finite values of w at branch-points. For the infinities of w, which may 
arise in connection with the singularities of the coefficients, we take the zeros of the 
coefficient of the highest power of w in the integral equation, viz., 2 = 0, which is thus the 
only infinity of w. To find its order we take w=z~ n f (z)yz~ n + ...... , where y is a 

constant and f (z) is finite for 2 = 0; and then we have 

8z l ~ 3n 



J. "~ 

Thus l-3n=-n, 

provided both of them be negative; the equality gives n = \ and satisfies the condition. 
And 8y 3 = - 3y. Of these values one is zero, and gives a branch of the function without 
an infinity; the other two are ^V-f and they give the initial term of the two 
branches of w, which have an infinity of order -^ at the origin and are cyclically 
interchangeable for a small circuit round it. The three values of w for infinitesimal 
values of z are 

3 . _i 1 7 /3 . l 4 275 
" - 81 -1944 



3 - 



M + /?&* 1*4. 2 1 5 /3-f_jL2_ 

6 18 V 8 81 1944 V 8 729 z 



_ _i A As 
w 3 --g + gj2+ 2 + 

The first two of these form the system for the branch-point at the origin, which is neither 
an infinity nor a critical point for the third branch of the function. 

Ex. 2. Obtain the branch-points of the functions which are defined by the following 
equations, and determine the cyclical systems at the branch-points : 
(i) w* 
(ii) w 
(iii) w 
(iv) iff 



44 
(v) vfi - (1 - a 2 ) 104 _ _ Z 2 (! _ 2 2)4 = 0- ( Briot and Bouquet.) 

Also discuss the branches, in the vicinity of 2 = and of 2=00, of the functions defined 
by the following equations : 

(vi) aw 7 + buz + cutz* + dwW + ewz 1 +fz 9 + gv + hw* + kz w = ; 
(vii) w m z n +w n +z m = Q. 



174 SIMPLE BRANCH-POINTS [98. 

98. There is one case of considerable importance which, though limited 
in character, is made the basis of Clebsch and Gordan s investigations* in the 
theory of Abelian functions the results being, of course, restricted by the 
initial limitations. It is assumed that all the branch-points are simple, that 
is, are such that only one pair of branches of w are interchanged by a circuit 
of the variable round the point ; and it is assumed that the equation /= is 
algebraical not merely in w but also in z. The equation f = can then be 
regarded as the generalised form of the equation of a curve of the nth order, 
the generalisation consisting in replacing the usual coordinates by complex 
variables; and it is further assumed, in order to simplify the analysis, that all 
the multiple points on the curve are (real or imaginary) double-points. But, 
even with the limitations, the results are of great value : and it is therefore 
desirable to establish the results that belong to the present section of the 
subject. 

We assume, therefore, that the branch-points are such that only one 
pair of branches of w are interchanged by a small closed circuit round any 
one of the points. The branch-points are among the values of z determined 
by the equations 

z) A 
> 



When /=0 has the most general form consistent with the assigned 
limitations, f (w, z) is of the ?ith degree in z ; the values of z are determined 
by the eliminant of the two equations which is of degree n(n 1), and there 
are, therefore, n(n Y) values of z which must be examined. 

First, suppose that J \, does not vanish for a value of z, thus 

oz 

obtained, and the corresponding value of w : then we have the first case 
in the preceding investigation. And, on the hypothesis adopted in the 
present instance, m = 2 ; so that each such point z is a branch-point. 

Next, suppose that ^ - vanishes for some of the n(n 1) values of z ; 

the value of m is still 2, owing to the hypothesis. The case will now be still 

d 2 f (w z} 
further limited by assuming that ^ . 2 does not vanish for the value of z 

and the corresponding value of w ; and thus in the vicinity of z = a, w = a we 
have an equation 

= Az- + 2Bz w + Cw 2 -f terms of the third degree + ...... , 

where A, B, C are the values of ^ , =-- , ~ f r z a > w=a. 

oz 1 dzdw ow 2 

If B 2 AC, this equation leads to the solution 

C w + Bz oc uniform function of z. 

* Clebsch und Gordan, Theorie der AbeVschen Functionen, (Leipzig, Teubner, 1866). 



98.] SIMPLE BRANCH-POINTS 175 

The point z = a, w = a is not a branch-point ; the values of w, equal at the 
point, are functionally distinct. Moreover, such a point z occurs doubly in 
the eliminant; so that, if there be B such points, they account for 28 in 
the eliminant of degree n (n 1) ; and therefore, on their score, the number 
n (n 1) must be diminished by 28. The case is, reverting to the genera 
lisation of the geometry, that of a double point where the tangents are 
not coincident. 

If, however, B 2 = AC, the equation leads to the solution 

Cw + Bz = Lz ^ + Mz * + Nz * + 

The point z = a, w = a is a point where the two values of z interchange. 
Now such a point z occurs triply in the eliminant ; so that, if there be K 
such points, they account for SK of the degree of the equation. Each of 
them provides only one branch-point, and the aggregate therefore provides K 
branch-points ; hence, in counting the branch-points of this type as derived 
through the eliminant, its degree must be diminished by 2/c. The case is, 
reverting to the generalisation of the geometry, that of a double point (real 
or imaginary) where the tangents are coincident. 

It is assumed that all the n(n 1) points z are accounted for under 
the three classes considered. Hence the number of branch-points of the 
equation is 

l = n (n - 1) - 28 - 2, 

where n is the degree of the equation, B is the number of double points 
(in the generalised geometrical sense) at which tangents to the curve do not 
coincide, and K is the number of double points at which tangents to the 
curve do coincide. 

And at each of these branch-points, II in number, two branches of the 
function are equal and, for a small circuit round it, interchange. 

99. The following theorem is a combined converse of many of the 
theorems which have been proved : 

A function w, which has n (and only ?/) values for each value of z, and 
which has a finite number of infinities and of branch-points in any part of the 
plane, is a root of an equation in w of degree n, the coefficients of which are 
uniform functions of z in that part of the plane. 

We shall first prove that every integral symmetric function of the n 
values is a uniform function in the part of the plane under consideration. 

n 

Let Sk denote 2, w, where k is a positive integer. At an ordinary point 

i-\ 

of the plane, S k is evidently a one-valued function and that value is finite ; 
S k is continuous ; and therefore the function S k is uniform in the immediate 
vicinity of an ordinary point of the plane. 



176 FUNCTIONS POSSESSING [99. 

For a point a, which is a branch-point of the function w, we know that 
the branches can be arranged in cyclical systems. Let w 1 ,..., w^ be such a 
system. Then these branches interchange in cyclical order for a description 
of a small circuit round a ; and, if z a = Z*, it is known ( 93) that, in the 
vicinity of Z = 0, a branch w is a, uniform function of Z, say 



Therefore w k = G k ) + P k (Z) 

\il 

in the vicinity of Z = ; say 

w* = A k + 2B k>m Z- + 2 C k>m Z. 

m=l m=l 

Now the other branches of the function which are equal at a are derivable 
from any one of them by taking the successive values which that one 
acquires as the variable describes successive circuits round a. A circuit 
of w round a changes the argument of z a, by 27r. and therefore gives Z 
reproduced but multiplied by a factor which is a primitive /xth root of unity, 
say by a factor a ; a second circuit will reproduce Z with a factor a 2 ; and so 
on. Hence 

wf = A k +2 B k>m a Z- +ZC k>m a- # 



w r k = A k +2 B k>m a- Z~ m + 2 C k , m a Z m , 

m=l =! 







and therefore 
I* 





w r k = pA k + 2 B km - + ar + cr + . . . + cr t 

r=l m = 1 

+ 2 flto* Z m (1 + m + a 2 "* + + a""*-). 

OT = 1 

Now, since a is a primitive /*th root of unity, 

1 +a s + 2S + ... + a s( x - 1) 

is zero for all integral values of s which are not integral multiples of p,, and it 
is yu, for those values of s which are integral values of jj, ; hence 

- 



B k> i(z - a)" 1 + B k ^(z - a)~ 2 + B kt3 (z - a) 



. 
Hence the point z = a may be a singularity of 2 w r k but it is not a branch- 

r=l 



99.] A FINITE NUMBER OF BRANCHES 177 

point of the function ; and therefore in the immediate vicinity of z a the 

*i 
quantity X w r k is a uniform function. 



r=l 



The point a is an essential singularity of this uniform function, if the 
order of the infinity of w at a be infinite : it is an accidental singularity, if 
that order be a finite integer. 

This result is evidently valid for all the cyclical systems at a, as well as 
for the individual branches which may happen to be one-valued at a. Hence 

(U. 

Sk, being the sum of sums of the form 2) w r k each of which is a uniform 

r=l 

function of z in the vicinity of a, is itself a uniform function of z in that 
vicinity. Also a is an essential singularity of Sk, if the order of the infinity at 
z = a for any one of the branches of w be infinite ; and it is an accidental 
singularity of S k> if the order of the infinity at z = a for all the branches of w 
be finite. Lastly, it is an ordinary point of Sk, if there be no branch of w for 
which it is an infinity. Similarly for each of the branch-points. 

Again, let c be a regular singularity of any one (or more) of the branches 
of w ; then c is a regular singularity of every power of each of those branches, 
the singularities being simultaneously accidental or simultaneously essential. 
Hence c is a singularity of 8k : and therefore in the vicinity of c, $& is a 
uniform function, having c for an accidental singularity if it be so for each of 
the branches w affected by it, and having c for an essential singularity if it be 
so for any one of the branches w. 

It thus appears that in the part of the plane under consideration the 
function 8k is one-valued ; and it is continuous and finite, except at certain 
isolated points each of which is a singularity. It is therefore a uniform 
function in that part of the plane ; and the singularity of the function at any 
point is essential, if the order of the infinity for any one of the branches w at 
that point be infinite, but it is accidental, if the order of the infinity for all the 
branches w there be finite. And the number of these singularities is finite, 
being not greater than the combined number of the infinities of the function 
w, whether regular singularities or branch-points. 

Since the sums of the kth powers for all positive values of the integer k 
are uniform functions and since any integral symmetric function of the n 
values is a rational integral algebraical function of the sums of the powers, it 
follows that any integral symmetric function of the n values is a uniform 
function of z in the part of the plane under consideration ; and every infinity 
of a branch w leads to a singularity of the symmetric function, which is 
essential or accidental according as the orders of infinity of the various 
branches are not all finite or are all finite. 

F. 12 



178 FUNCTIONS POSSESSING [99. 

Since w has n (and only n) values w lt ... ,w n for each value of z, the 
equation which determines w is 

(W - Wj) (W-W 2 ) ... (W- W n ) = 0. 

The coefficients of the various powers of w are symmetric functions of the 
branches w l , . . . , w n ; and therefore they are uniform functions of z in the 
part of the plane under consideration. They possess a finite number of 
singularities, which are accidental or essential according to the character of 
the infinities of the branches at the same points. 

COROLLARY. If all the infinities of the branches in the finite part of the 
whole plane be of finite order, then the finite singularities of all the coefficients 
of the powers of w in the equation satisfied by w are all accidental ; and the 
coefficients themselves then take the form of a quotient of an integral uniform 
function (which may be either transcendental or algebraical, in the sense of 
47) by another function of a similar character. 

If z = oc be an essential singularity for at least one of the coefficients, 
through being an infinity of unlimited order for a branch of w, then one 
or both of the functions in the quotient-form of one at least of the coefficients 
must be transcendental. 

If z = oo be an accidental singularity or an ordinary point for all the 
coefficients, through being either an infinity of finite order or an ordinary 
point for the branches of w, then all the functions which occur in all the 
coefficients are rational, algebraical expressions. When the equation is 
multiplied throughout by the least common multiple of the denominators 
of the coefficients, it takes the form 

w n h (z) + w n ~* A, (z) + . . . + w h n _, (z} + h n (z) = 0, 

where the functions h (z), h^(z\ ..., h n (z) are rational, integral, algebraical 
functions of z, in the sense of 47. 

A knowledge of the number of infinities of w gives an upper limit of the 
degree of the equation in z in the last form. Thus, let a t be a regular 
singularity of the function ; and let Oi, fa, ji, ... be the orders of the infinities 
of the branches at a t - ; then 

w^w-i ... w n (z a t ) A , 
where \ denotes Oi + fti + % + ..., is finite (but not zero) for z = a t . 

Let Ci be a branch-point, which is an infinity; and let p, branches w form a 

ft 
system for c t -, such that w(z Cf)^ is finite (but not zero) at the point; then 

w : w 2 ... Wp (z Q) 
is finite (but not zero) at the point, and therefore also 



99.] A FINITE NUMBER OF BRANCHES 179 

is finite, where Q it (/>;, ^i, ... are numbers belonging to the various systems; 
or, if ei denote 0; + $f + tyi + . . . , then 

W l ...W n (z- Ci) 6i 

is finite for z = C;. Similarly for other symmetric functions of w. 

Hence, if j, a 2 , ... be the regular singularities with numbers X 1; X 2 , ... 
defined as above, and if c^ c 2 , ... be the branch -points, that are also infinities, 
with numbers e 1; e 2 , ... defined as above, then the product 

(w-Wj) ...... (w-w n ) n 0-a f ) A< n 0-Ci) e< 

i=l 1=1 

is finite at all the points ai and at all the points c;. The points a and the 
points c are the only points in the finite part of the plane that can make the 
product infinite : hence it is finite everywhere in the finite part of the plane, 
and it is therefore an integral function of z. 

Lastly, let p be the number for z = oo corresponding to \i for a f or to e^ 
for C;, so that for the coefficient of any power of w in (w w^) ...(w w n ) the 
greatest difference in degree between the numerator and the denominator is 
p in favour of the excess of the former. 

Then the preceding product is of order 



which is therefore the order of the equation in z when it is expressed in a 
holomorphic form. 



122 



CHAPTER IX. 

PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS IN GENERAL. 

100. INSTANCES have already occurred in which the value of a function 
of z is not dependent solely upon the value of z but depends also on the 
course of variation by which z obtains that value ; for example, integrals of 
uniform functions, and multiform functions. And it may be expected that, 
a fortiori, the value of an integral connected with a multiform function will 
depend upon the course of variation of the variable z. Now as integrals 
which arise in this way through multiform functions and, generally, integrals 
connected with differential equations are a fruitful source of new functions, 
it is desirable that the effects on the value of an integral caused by variations 
of a -path be assigned so that, within the limits of algebraic possibility, the 
expression of the integral may be made completely determinate. 

There are two methods which, more easily than others, secure this result ; 
one of them is substantially due to Cauchy, the other to Riemann. 

The consideration of Riemann s method, both for multiform functions and 
for integrals of such functions, will be undertaken later, in Chapters XV., 
XVI. Cauchy s method has already been used in preceding sections relating 
to uniform functions, and it can be extended to multiform functions. Its 
characteristic feature is the isolation of critical points, whether regular 
singularities or branch-points, by means of small curves each containing one 
and only one critical point. 

Over the rest of the plane the variable z ranges freely and, under certain 
conditions, any path of variation of z from one point to another can, as will 
be proved immediately, be deformed without causing any change in the 
value of the integral, provided that the path does not meet any of the small 
curves in the course of the deformation. Further, from a knowledge of the 
relation of any point thus isolated to the function, it is possible to calculate 
the change caused by a deformation of the -path over such a point; and 
thus, for defined deformations, the value of the integral can be assigned 
precisely. 



100.] INTEGRAL OF A BRANCH 181 

The properties proved in Chapter II. are useful in the consideration of 
the integrals of uniform functions ; it is now necessary to establish the 
propositions which give the effects of deformation of path on the integrals 
of multiform function. The most important of these propositions is the 
following : 

f b 
If w be a multiform function, the value of I wdz, taken between two 

J a 

ordinary points, is unaltered for a deformation of the path, provided that the 
initial branch of w be the same and that no branch-point or infinity be crossed 
in the deformation. 

Consider two paths acb, adb, (fig. 16, p. 152), satisfying the conditions 
specified in the proposition. Then in the area between them the branch w 
has no infinity and no point of discontinuity ; and there is no branch-point 
in that area. Hence, by 90, Corollary VI., the branch w is a uniform 
monogenic function for that area; it is continuous and finite everywhere 
within it and, by the same Corollary, we may treat w as a uniform, mono 
genic, finite and continuous function. Hence, by 17, we have 

rb ra 

(c) I wdz + (d) wdz = 0, 

J a J b 

the first integral being taken along acb and the second along bda; and 
therefore 

rb ra rb 

(c) wdz = (d}\ wdz = (d) \ wdz, 

Jo, J b J a 

shewing that the values of the integral along the two paths are the same 
under the specified conditions. 

It is evident that, if some critical point be crossed in the deformation, 
the branch w cannot be declared uniform and finite in the area and the 
theorem of 17 cannot then be applied. 

COROLLARY I. The integral round a closed curve containing no critical 
point is zero. 

COROLLARY II. A curve round a branch-point, containing no other 
critical point of the function, can be deformed into a loop 
without altering the value of fwdz ; for the deformation 
satisfies the condition of the proposition. Hence, when 
the value of the integral for the loop is known, the 
value of the integral is known for the curve. 

COROLLARY III. From the proposition it is possible 
to infer conditions, under which the integral fwdz round 
the whole of any curve remains unchanged, when the whole 
curve is deformed, without leaving an infinitesimal arc 
common as in Corollary II. 





182 INTEGRATION [100. 

Let GDC , ABA be the curves: join two consecutive points A A to two 
consecutive points (7(7. Then if the area CABA C DG 
enclose no critical point of the function w, the value of 
jwdz along CDC is by the proposition the same as its 
value along CABA C . The latter is made up of the 
value along CA, the value along ABA , and the value 
along AC , say 

rA r rC 

I wdz + I wdz + w dz, v . 

Jc JB JA . 

where w is the changed value of w consequent on the description of a simple 
curve reducible to B ( 90, Cor. II.). 

Now since w is finite everywhere, the difference between the values of w 
at A and at A consequent on the description of ABA is finite : hence as 
A A is infinitesimal the value of jwdz necessary to complete the value for 
the whole curve B is infinitesimal and therefore the complete value can be 

taken as the foregoing integral wdz. Similarly for the complete value 

J B 

along the curve D : and therefore the difference of the integrals round B and 
round D is 

rA rC 

I wdz + I w dz, 

J C J A 

rA 

say (w w ) dz. 

J c 

In general this integral is not zero, so that the values of the integral 
round B and round D are not equal to one another : and therefore the curve 
D cannot be deformed into the curve B without affecting the value of jwdz 
round the whole curve, even when the deformation does not cause the curve 
to pass over a critical point of the function. 

But in special cases it may vanish. The most important and, as a 
matter of fact, the one of most frequent occurrence is that in which the 
description of the curve B restores at A the initial value of w at A. It 
easily follows, by the use of 90, Cor. II., that the description of D (as 
suming that the area between B and D includes no critical point) restores 
at C the initial value of w at (7. In such a case, w = w for corresponding 
points on AC and A C , and the integral, which expresses the difference, is 
zero: the value of the integral for the curve B is then the same as that for D. 
Hence we have the proposition : 

If a curve be such that the description of it by the independent variable 
restores the initial value of a multiform function w, then the value of jwdz 
taken round the curve is unaltered when the curve is deformed into any other 
curve, provided that no branch-point or point of discontinuity of w is crossed 
in the course of deformation. 



100.] OF MULTIFORM FUNCTIONS 183 

This is the generalisation of the proposition of 19 which has thus far 
been used only for uniform functions. 

Note. Two particular cases, which are very simple, may be mentioned 
here : special examples will be given immediately. 

The first is that in which the curve B, and therefore also D, encloses 
no branch-point or infinity; the initial value of w is restored after a 
description of either curve, and it is easy to see (by reducing B to a 
point, as may be done) that the value of the integral is zero. 

The second is that in which the curve encloses more than one branch 
point, the enclosed branch-points being such that a circuit of all the loops, 
into which (by Corollary V., 90) the curve can be deformed, restores the 
initial branch of w. This case is of especial importance when w is two-valued : 
the curves then enclose an even number of branch-points. 

101. It is important to know the value of the integral of a multiform 
function round a small curve enclosing a branch-point. 

Let c be a point at which TO branches of an algebraical function are equal 
and interchange in a single cycle ; and let c, if an infinity, be of only finite 
order, say k/m. Then in the vicinity of c, any of the branches w can be 
expressed in the form 

00 . 

w= 2 g s (z-c) m , 

o If 

o K 

where k is a finite integer. 

The value of jwdz taken round a small curve enclosing c is the sum of 
the integrals 



the value of which, taken once round the curve and beginning at a point z ly is 



TO + S 

where a is a primitive mth root of unity, provided TO + s is not zero. If then 
s + m be positive, the value is zero in the limit when the curve is infini 
tesimal : if TO + s be negative, the value is oo in the limit. 

But, if m + s be zero, the value is Z7rig s . 

Hence we have the proposition: If, in the vicinity of a branch-point c, 
where m branches w are equal to one another and interchange cyclically, the 
expression of one of the branches be 



184 MULTIPLICITY OF VALUE [101. 

then jwdz, taken once round a small curve enclosing c, is zero, if k<m; is 
infinite, if k> m ; and is ^irig^ , if k = m. 

It is easy to see that, if the integral be taken m times round the small 
curve enclosing c, then the value of the integral is 2m7rig m when k is greater 
than in, so that the integral vanishes unless there be a term involving (z c)" 1 
in the expansion of a branch w in the vicinity of the point. The reason that 
the integral, which can furnish an infinite value for a single circuit, ceases to 

_* 

do so for m circuits, is that the quantity (^ c) m , which becomes indefi 
nitely great in the limit, is multiplied for a single circuit by a* 1, for a 
second circuit by a 2A a A , and so on, and for the mth circuit by a wA a (w ~ 1)A , 
the sum of all of which coefficients is zero. 

Ex. The integral \{(z - a) (z - b) ... (z -f)}~* dz taken round an indefinitely small curve 
enclosing a is zero, provided no one of the quantities b, ... ,/ is equal to a. 

102. Some illustrations have already been given in Chapter II., but 
they relate solely to definite, not to indefinite, integrals of uniform 
functions. The whole theory will not be considered at this stage ; we shall 
merely give some additional illustrations, which will shew how the method 
can be applied to indefinite integrals of uniform functions and to integrals 
of multiform functions, and which will also form a simple and convenient 
introduction to the theory of periodic functions of a single variable. 

We shall first consider indefinite integrals of uniform functions. 



f dz 
Ex. 1. Consider the integral I , and denote* it by/ (z}. 



The function to be integrated is uniform, and it has an accidental singularity of the first 
order at the origin, which is its only singularity. The value of \z~ l dz taken positively 
along a small curve round the origin, say round a circle with the origin as centre, is 2n-i 
but the value of the integral is zero when taken along any closed curve which does not 
include the origin. 

Taking z = l as the lower limit of the integral, and any point z as the upper limit, we 
consider the possible paths from 1 to z. Any path from 1 to z can be deformed, without 
crossing the origin, into a path which circumscribes the origin positively some number of 
times, say m^, and negatively some number of times, say i 2 , all in any order, and then leads 
in a straight line from 1 to z. For this path the value of the integral is equal to 



I , 
J 1 z 

that is, to 2mni+ I , 

Ji z 

where m is an integer, and in the last integral the variation of z is along a straight 
line from 1 to z. Let the last integral be denoted by u ; then 



* See Chrystal, ii, pp. 266 272, for the elementary properties of the function and its inverse, 
when the variable is complex. 



102.] OF INTEGRALS 185 

and therefore, inverting the function and denoting/" 1 by <j>, we have 



Hence the general integral is a function of z with an infinite number of values ; and z is a 
periodic function of the integral, the period being 2n-z. 

Ex. 2. Consider the function / - - ^ > an d again denote it by / (z). 

The one- valued function to be integrated has two accidental singularities + i, each of 
the first order. The value of the integral taken positively along a small curve round i is 
TT, and along a small curve round i is n. 

We take the origin as the lower limit and any point z as the upper limit. Any path 
from to z can be deformed, without crossing either of the singularities and therefore 
without changing the value of the integral, into 

(i) any numbers of positive (m l5 w? 2 ) an( * of negative (nz/, m 2 ) circuits round i and 
round -i, and 

(ii) a straight line from to z. 
Then we have 

- TJ-) +WIJJ ( - IT) + m. 2 {_(-)}+ /* . 

J o 

, z 
= nir+ 



where ?i is an integer and the integral on the right-hand side is taken along a straight line 
from to z. 

Inverting the function and denoting/" 1 by tp, we have 



The integral, as before, is a function of z with an infinite number of values ; and z is a 
periodic function of the integral, the period being TT. 

103. Before passing to the integrals of multiform functions, it is con 
venient to consider the method in which Hermite* discusses the multiplicity 
in value of a definite integral of a uniform function. 

Taking a simple case, let <> (X) = \ 

J Q 1 + Z 

and introduce a new variable t such that Zzt\ then 

zdt 

When the path of t is assigned, the integral is definite, finite and unique in 
value for all points of the plane except for those for which 1 + zt = ; and, 
according to the path of variation of t from to 1, there will be a 0-curve 
which is a curve of discontinuity for the subject of integration. Suppose the 
path of t to be the straight line from to 1 ; then the curve of discontinuity 

* Crelle, t. xci, (1881), pp. 6277; Cours a la Faculte des Sciences, 4 6me 6d. (1891), pp. 
7679, 154164, and elsewhere. 



186 HERMITE S [103. 

is the axis of x between 1 and oo . In this curve let any point - be 
taken where > 1 ; and consider a point z 1 -^ + ie and a point z 2 = ie, 
respectively on the positive and the negative sides of the axis of x, both 
being ultimately taken as infinitesimally near the point . Then 



dt= ( 



Let e become infinitesimal ; then, when t is infinite, we have 



tan 



for e is positive ; and, when t is unity, we have 



tan" 1 ----- = |TT, 



for is > 1. Hence < (^) < (^ 2 ) 

The part of the axis of x from - 1 to - oo is therefore a line of discon 
tinuity in value of <j> (z), such that there is a sudden change in passing from 
one edge of it to the other. If the plane be cut along this line so that 
it cannot be crossed by the variable which may not pass out of the plane, 
then the integral is everywhere finite and uniform in the modified surface. 
If the plane be not cut along the line, it is evident that a single passage 
across the line from one edge to the other makes a difference of 2?ri in the 
value, and consequently any number of passages across will give rise to the 
multiplicity in value of the integral. 

Such a line is called a section* by Hermite, after Riemann who, in a 
different manner, introduces these lines of singularity into his method of 
representing the variable on surfaces "f*. 

When we take the general integral of a uniform function of Z and make 
the substitution Z = zt, the integral that arises for consideration is of the form 



We shall suppose that the path of variation of t is the axis of real quantities : 
and the subject of integration will be taken to be a general function of t and 
z, without special regard to its derivation from a uniform function of Z. 

* Coupure; see Crelle, t. xci, p. 62. t See Chapter XV. 



103.] SECTIONS 187 

It is easy, after the special example, to see that ^ is a continuous function 
of z in any space that does not include a ^-point which, for values of t included 
within the range of integration, would satisfy the equation. 

G (t, z) = 0. 

But in the vicinity of a ^-point, say , corresponding to the value t = 6 in 
the range of integration, there will be discontinuity in the subject of 
integration and also, as will now be proved, in the value of the integral. 

Let Z be the point and draw the curve through Z corresponding to 
t = real constant ; let N t be a point on the positive side and N 2 
a point on the negative side of this curve positively described, 
both points being on the normal at Z ; and let 
supposed small. Then for N! we have 

X-L = g e sin y, y l = ^-\-e cos y , 

Fig. 24. 
so that z 1 = +16 (cosy + isiny), 

where ty is the inclination of the tangent to the axis of real quantities. But, 
if da- be an arc of the curve at Z, 

da , i \ d% dt] d 

for variations along the tangent at Z, that is, 

i 

da- . . 3 

-j-- (cos y + i sin y ) = - 




Thus, since -j- may be taken as finite on the supposition that Z is an 
ordinary point of the curve, we have 



where e = e -y- , P = - 

Similarly z. 2 = + ie -^r. 

Hence <1> (^) = I --i-i * ^ 

w/ n_^J_w/ m _ 

1*. 



188 HERMITE S [103. 

with a similar expression for <& (z 2 ) ; and therefore 

F(t, j- [G (t, }^-G (t, ) 





The subject of integration is infinitesimal, except in the immediate vicinity 
of t = 6 ; and there 



powers of small quantities other than those retained being negligible. Let 
the limiting values of t, that need be retained, be denoted by d + v and 
d p , then, after reduction, we have 

edt 



F(e, 



in the limit when e is made infinitesimal. 

Hence a line of discontinuity of the subject of integration is a section 
for the integral ; and the preceding expression is the magnitude, by 
numerical multiples of which the values of the integral differ*. 

Ex. 1. Consider the integral 

dZ 



/ 



zdt 
h 

We have S ^ * =^ = ^g = ^. 



so that TT is the period for the above integral. 
Ex. 2. Shew that the sections for the integral 



t a sin z , 
2 



* The memoir and the Cmirs d Analyse of Hermite should be consulted for further develop 
ments; and, in reference to the integral treated above, Jordan, Cours d Analyse, t. iii, pp. 
610 614, may be consulted with advantage. See also, generally, for functions defined by 
definite integrals, Goursat, Acta Math., t. ii, (1883), pp. 170, and ib., t. v, (1884), pp. 97 
120; and Pochhammer, Math. Arm., t. xxxv, (1890), pp. 470494, 495526. Goursat also 
discusses double integrals. 



103.] SECTIONS 189 



where a is positive and less than 1, are the straight lines x = (2k + l) TT, where k assumes all 
integral values ; and that the period of the integral at any section at a distance 77 from the 
axis of real quantities is 2?r cosh (arj). (Hermite.) 

Ex. 3. Shew that the integral 



o 

where the real parts of /3 and y /3 are positive, has the part of the axis of real quantities 
between 1 and +00 for a section. 

Shew also that the integral 

i 

rht } ( z P~ I (~i - v y ~ 3 ~ 1 n }~ a d 

J 

where the real parts of /3 and 1 - a are positive, has the part of the axis of real quantities 
between and 1 for a section : but that, in order to render < (z) a uniform function of z, 
it is necessary to prevent the variable from crossing, not merely the section, but also the 
part of the axis of real quantities between 1 and + <x> . (Goursat.) 

(The latter line is called a section of the second kind.) 

Ex. 4. Discuss generally the effect of changing the path of t on a section of the 

integral ; and, in particular, obtain the section for I when, after the substitution 

jo 1 + 

Z=zt, the path of t is made a semi-circle on the line joining and 1 as diameter. 

Note. It is manifestly impossible to discuss all the important bearings of theorems and 
principles, which arise from time to time in our subject ; we can do no more than mention 
the subject of those definite integrals involving complex variables, which first occur as 
solutions of the better-known linear differential equations of the second order. 

Thus for the definite integral connected with the hypergeometric series, memoirs by 
Jacobi* and Goursat t should be consulted ; for the definite integral connected with 
Bessel s functions, memoirs by HankelJ and Weber should be consulted ; and Heine s 
J/andbuch der Kugelfunctionen for the definite integrals connected with Legendre s 
functions. 

104. We shall now consider integrals of multiform functions. 

Ex. 1. To find the integral of a multiform function round one loop ; and round a 
number of loops. 

Let the function be 

i 

w={(z-a l }(z-a. z }...(z- a n )} , 

where m may be a negative or positive integer, and the quantities a are unequal to one 
another ; and let the loop be from the origin round the point a x . Then, if / be the value 
of the integral with an assigned initial branch w, we have 



/a, f CO 

wdz-\- I wdz + I awdz, 
J c J a. 



where a is e m and the middle integral is taken round the circle at a^ of infinitesimal radius. 

* Crelle, t. Ivi, (1859), pp. 149 165 ; the memoir was not published until after his death, 
t Sur Vequation differentielle lineaire qui admet pour integrate la serie hypergrometrique, 
(These, Gauthier-Villars, Paris, 1881). 

I Math. Ann., t. i, (1869), pp. 467501. 

Math. Ann., t. xxxvii, (1890), pp. 404416. 



190 



EXAMPLES 



[104. 



But, since the limit of (z-ajw when z = a 1 is zero, the middle integral vanishes by 101 ; 
and therefore 



/"i 
, = (! -a) I web, 

Jo 



where the integral may, if convenient, be considered as taken along the straight line from 
to a l . 






(2) 



(3) 



Fig. 25. 



Next, consider a circuit for an integral of w which (fig. 25) encloses two branch-points, 
say ! and 2 , but no others ; the circuit in (1) can be deformed into that in (2) or into 
that in (3) as well as into other forms. Hence the integral round all the three circuits 
must be the same. Beginning with the same branch as in the first case, we have 



(1 



/! 
wdz, 
o 



as the integral after the first loop in (2). And the branch with which the second loop 
begins is aw, so that the integral described as in the second loop is 

/2 
awdz; 


and therefore, for the circuit as in (2), the integral is 

Ca t [a y 

1= (1 - a) I wdz + a (1 - a) / wdz. 
Jo Jo 

Proceeding similarly with the integral for the circuit in (3), we find that its expression is 

/a 2 /"<*! 

wdz + a (I -a) I wdz, 
J 

and these two values must be equal. 

But the integrals denoted by the same symbols are not the same in the two cases ; the 

function I * wdz is different in the second value of J from that in the first, for the deforma- 

Jo 

tion of path necessary to change from the one to the other passes over the branch-point a z . 
In fact, the equality of the two values of / really determines the value of the integral for 
the loop Oa l in (3). 

And, in general, equations thus obtained by varied deformations do not give relations 
among loop-integrals but define the values of those loop-integrals for the deformed paths. 

We therefore take that deformation of the circuit into loops which gives the simplest 
path. Usually the path is changed into a group of loops round the branch-points as they 
occur, taken in order in a trigonometrically positive direction. 

The value of the integral round a circuit, equivalent to any number of loops, is obvious. 

Ex. 2. To find the value of $wdz, taken round a simple curve which includes all the 
branch-points of w and all the infinities. 



104.] OF PERIODICITY OF INTEGRALS 191 

If z = oo be a branch-point or an infinity, then all the branch-points and all the 
infinities of w lie on what is usually regarded as the exterior of the curve, or the curve 
may in one sense be said to exclude all these points. The integral round the curve is then 
the integral of a function round a curve, such that over the area included by it the 
function is uniform, finite and continuous ; hence the integral is zero. 

If = 00 be neither a branch-point nor an infinity, the curve can be deformed until it is 
a circle, centre the origin and of very great radius. If then the limit of zw, when \z is 
infinitely great, be zero, the value of the integral again is zero, by II., 24. 

Another method of considering the integral, is to use Neumann s sphere for the 
representation of the variable. Any simple closed curve divides the area of the sphere 
into two parts ; when the curve is defined as above, one of those parts is such that the 
function is uniform, finite and continuous throughout and therefore its integral round the 
curve, regarded as the boundary of that part, is zero. (See Corollary III., 90.) 

Ex. 3. To find the general value of J(l-2 2 )~*cfe. The function to be integrated is 
two-valued: the two values interchange round each of the branch-points 1, which are 
the only branch-points of the function. 

Let / be the value of the integral for a loop from the origin round +1, beginning with 
the branch which has the value +1 at the origin ; and let / be the corresponding value 
for the loop from the origin round - 1, beginning with the same branch. Then, by Ex. 1, 

/= 2 P (1 - z*T*dz, / = 2 f" 1 (1 - z 2 )"* dz 

= -/, 

the last equality being easily obtained by changing variables. 

Now consider the integral when taken round a circle, centre the origin and of indefinitely 
great radius R ; then by 24, II., if the limit of zw for z= QO be k, the value of \wdz round 

this circle is 2iri&. In the present case w = (l- 2 2 )~^ so that the limit of zw is + ^ ; hence 

J(l-2 2 r^2 = 27T, 

the integral being taken round the circle. But since a description of the circle restores the 

initial value, it can be deformed into the two loops from O 

to A and from to A . The value round the first is /; and ^ r > ^ 

the branch with which the second begins to be described has 

the value 1 at the origin, so that the consequent value round * 1 S- ^"- 

the second is / ; hence 

7-/ = 2r* 

and therefore 

verifying the ordinary result that 



when the integral is taken along a straight line. 

To find the general value of u for any path of variation between and z, we proceed as 
follows. Let Q be any circuit which restores the initial branch of (l-z 2 )~^. Then by 
100, Corollary II., Q may be composed of 

(i) a set of double circuits round + 1, say m , 
(ii) a set of double circuits round - 1, say m", 
and (iii) a set of circuits round + 1 and - 1 ; 

* It is interesting to obtain this equation when O is taken as the initial point, instead of 0. 



192 EXAMPLES OF PERIODICITY [104. 

and these may come in any order and each may be described in either direction. Now for 
a double circuit positively described, the value of the integral for the first description is / 
and for the second description, which begins with the branch (1 z 2 )~^, it is /; hence 
for the double circuit it is zero when positively described, and therefore it is zero also when 
negatively described. Hence each of the TO double circuits yields zero as its nett contribu 
tion to the integral. 

Similarly, each of the m" double circuits round - 1 yields zero as its nett contribution 
to the integral. 

For a circuit round + 1 and - 1 described positively, the value of the integral has just 
been proved to be /-/ , and therefore when described negatively it is / -/. Hence if 
there be n^ positive descriptions and n 2 negative descriptions, the nett contribution of all 
these circuits to the value of the integral is (n n^) (I - 1 ), that is, 2nir where n is an 
inteer. 




Hence the complete value for the circuit Q i 

Now any path from to z can be resolved into a circuit Q, which restores the initial 
branch of (1 2 2 )~ , chosen to have the value 
+ 1 at the origin, and either (i) a straight 
line Oz ; 

or (ii) the path OACz, viz., a loop round 
+ 1 and the line Oz ; 

or (iii) the path OA Cz, viz., a loop round 
- 1 and the line Oz. 

Let u denote the value for the line Oz, so that 

u= f* (!-#)-* dk. 

J o 

Hence, for case (i), the general value of the integral is 

2W7T + U. 

For the path OA Cz, the value is 7 for the loop OAC, and is ( u) for the line Cz, the 
negative sign occurring because, after the loop, the branch of the function for integra 
tion along the line is (1 2 2 )~ 5 ; this value is Iu, that is, it is TT U. Hence, for case 
(ii), the value of the integral is 

U. 



For the path OA Cz, the value is similarly found to be - TT - u ; and therefore, for case (iii), 
the value of the integral is 

2?wr ir-u. 

If /(z) denote the general value of the integral, we have either 



Or /(Z) = (2TO+1)7T-W, 

where n and m are any integers, so that/ (z) is a function with two infinite series of values. 
Lastly, if z = $($) be the inverse oif(z} = 6, then the relation between u and z given by 



can be represented in the form 
and 



104.] OF INTEGRALS 193 

both equations being necessary for the full representation. Evidently z is a simply -periodic 
function of u, the period being 2?r ; and from the definition it is easily seen to be an odd 
function. 

Let y = (\ -z 2 )x ( u \ so that y is an even function of u ; from the consideration of the 
various paths from to 2, it is easy to prove that 



Ex. 4. To find the general value of f{(l-j*)(l-IM)}~*dk It will be convenient 
(following Jordan *) to regard this integral as a special case of 

Z= \{(z -a)(z- b) (z -c}(z- d)}~* dz = \wdz. 

The two-valued function to be integrated has a, 6, c, d (but not oo ) as the complete 
system of branch-points ; and the two values interchange at each of them. We proceed as 
in the last example, omitting mere re-statements of reasons there given that are applicable 
also in the present example. 

Any circuit Q, which restores an initial branch of w, can be made up of 
(i) sets of double circuits round each of the branch-points, 
and (ii) sets of circuits round any two of the branch -points. 
The value of \wdz for a loop from the origin to a branch-point k (where k = a,b, c, or d) is 

2 I wdz ; 

J o 
and this may be denoted by K, where K=A, B, C, or D. 

The value of the integral for a double circuit round a branch-point is zero. Hence the 
amount contributed to the value of the integral by all the sets in (i) as this part of 
Q is zero. 

The value of the integral for a circuit round a and b taken positively is A - B ; for one 
round b and c is B- C ; for one round c and d is C-D; for one round a and c is A - C, 
which is the sum of A - B and B-C; and similarly for circuits round a and d and round 
b and d. There are therefore three distinct values, say A-B, B-C, C-D, the values 
for circuits round a and b, b and c, c and d respectively ; the values for circuits round any 
other pair can be expressed linearly in terms of these values. Suppose then that the part 
of Q represented by (ii), when thus resolved, is the nett equivalent of the description of m 
circuits round a and b, of n circuits round b and c, and of I circuits round c and d. Then 
the value of the integral contributed by this part of Q is 



which is therefore the whole value of the integral for Q. 

But the values of A, , C, D are not independent f. Let a circle with centre the origin 
and very great radius be drawn ; then since the limit of zw for |s| = oo is zero and since 
2= cc is not a branch-point, the value of \wdz round this circle is zero (Ex. 2). The circle 
can be deformed into four loops round a, b, c, d respectively in order ; and therefore the 
value of the integral is A - B + C- D, that is, 



Hence the value of the integral for the circuit fl is 



where m and n denote m - 1 and n - 1 respectively. 
* Cours d Analyse, t. ii, p. 343. 

t For a purely analytical proof of the following relation, see Greenhill s Elliptic Functions 
Chapter II. 

F - 13 



194 PERIODICITY [104. 

Now any path from the origin to z can be resolved into Q, together with either 
(i) a straight line from to z, 

or (ii) a loop round a and then a straight line to z. 

It might appear that another resolution would be given by a combination of Q with, say, a 
loop round b and then a straight line to z ; but it is resoluble into the second of the above 
combinations. For at C, after the description of the loop B , introduce a double description 
of the loop A, which adds nothing to the value of the integral and does not in the end 
affect the branch of w at C ; then the new path can be regarded as made up of (a) the 
circuit constituted by the loop round b and the first loop round a, (/3) the second loop round 
a, which begins with the initial branch of w, followed by a straight path to z. Of these 
(a) can be absorbed into G, and (/3) is the same as (ii) ; hence the path is not essentially 
new. Similarly for the other points. 

Let u denote the value of the integral with a straight path from to z; then the 
whole value of the integral for the combination of Q with (i) is of the form 



For the combination of O with (ii), the value of the integral for the part (ii) of 
the path is J, for the loop round a, +(-), for the straight path which, owing to the 
description of the loop round a, begins with - w ; hence the whole value of the integral is 
of the form 



Hence, if / (z) denote the general value of the integral, it has two systems of values, each 
containing a doubly -infinite number of terms; and, if z = <j>(u) denote the inverse of 
u = f (z\ we have 



= {m (A-B} + n(B-C)+A - u}, 

where m and n are any integers. Evidently z is a doubly-periodic function of u, with 
periods A-B and B-C. 

Ex. 5. The case of the foregoing integral which most frequently occurs is the elliptic 
integral in the form used by Legendre and Jacobi, viz. : 

u = J{(1 - z 2 ) (1 - kW)}-*dz = \wdz, 
where k is real. The branch-points of the function to be integrated are 1, -1, ^ 

and -L and the values of the integral for the corresponding loops from the origin are 

A/ 

A 
2 I wdz, 

J o 

r-i ri 

2 I wdz -2 I wdz, 
Jo / 

I wdz, 




and 

Now the values for the loops are connected by the equation 



* The value for a loop round b and then a straight line to z, just considered, is B - u 

= -(A-B) + 
being the value in the text with m changed to m - 1. 



104.] 



OF ELLIPTIC INTEGRALS 



195 



and so it will be convenient that, as all the points lie on the axis of real variables, we 
arrange the order of the loops so that this relation is identically satisfied. Otherwise, 
the relation will, after Ex. 1, be a definition of the paths of integration chosen for the 
loops. 

Among the methods of arrangement, which secure the identical satisfaction of the 



Fig. 28. 



relation, the two in the figure* are the simplest, the curved lines being taken straight in 
the limit ; for, by the first arrangement when k < 1, we have 





and, by the second when > 1, we have 



both of which are identically satisfied. We may therefore take either of them ; let the 
former be adopted. 

The periods are A-B, B-C, (and C-D, which is equal to B-A\ and any linear 
combination of these is a period: we shall take A - B, and B-D. The latter, B-D, 
is equal to 

n r-i 

2 / wdz -2 I wdz, 

Jo Jo 

which, being denoted by 4/f, gives 

4J5T=4 / 

JO{(1- 2 2)(1_222)}4 

as one period. The former, A-B, is equal to 

2 I wdz -2 I wdz, 
Jo Jo 

i 

/ wdz; 

/k 

1|(1- 



which is 2 

this, being denoted by 2iK , gives 



dz 



dz 



where 2 + 2 =l and the relation between the variables of the integrals is 

i 

Hence the periods of the integral are 4K and ZiK . Moreover, A is 2 I" wdz, which i 

i J 

2 / wdz + 2 I wdz = 
Jo J i 

Hence the general value of f* {(I - z*) (I - 



* Jordan, Cours d Analyse, t. ii, p. 356. 



132 



196 PERIODICITY [104. 



or 



that is, 2K-u + 4mK+2niK , 

where u is the integral taken from to z along an assigned path, often taken to be 
a straight line ; so that there are two systems of values for the integral, each containing 
a doubly -infinite number of terms. 

If z be denoted by $ (u) evidently, from the integral definition, an odd function 
of u , then 



so that z is a doubly-periodic function of u, the periods being 4A and 2iK . 

Now consider the function ^ = (1 -zrf. A 2-path round T does not affect ^ by way of 
change, provided the curve does not include the point 1 ; hence, if z t = x (u), we have 



But a z- path round the point 1 does change % into z 1 ; so that 

X ()--* (+**} 

Hence x ( u \ which is an even function, has two periods, viz., 4AT and 2A + 2i A", whence 

x (u) = x (u + 4mK+ 2nK+ 2niK ). 
Similarly, taking z 2 = (l -Fs 2 )* = -f (u), it is easy to see that 



so that ^ (u), which is an even function, has two periods, viz., 2 A and 4iK ; whence 



= u 



The functions < (u), x ( u \ ^ ( M ) are of course sn w > cn ""> dn M respectively. 
Ex. 6. To find the general value of the integral 



The function to be integrated has e^ e 2 , e 3 , and co for its branch-points; and for 
paths round each of them the two branches interchange. 

A circuit G which restores the initial branch of the function to be integrated can 
be resolved into : 

(i) Sets of double circuits round each of the branch-points alone : as before, the 

value of the integral for each of these double circuits is zero. 
(ii) Sets of circuits, each enclosing two of the branch-points : it is convenient to 
retain circuits including oo and e n oo and e. 2 , oc and e 3 , the other three 
combinations being reducible to these. 
The values of the integral for these three retained are respectively 

E! = 2 f (4 (z - ej (z - e 2 } (z - e^dz = 2 1 , 
J i 

E z =2 I {4(2-e 1 )(2-e 2 )(s-e 3 )}~ i ck=2a> 2 , 
J 62 

3 J e a 

* The choice of o> for the upper limit is made on a ground which will subsequently be 
considered, viz., that, when the integral is zero, z is infinite. 



104.] OF ELLIPTIC INTEGRALS 197 

and therefore the value of the integral for the circuit O is of the form 



But E^ K 2 , E 3 are not linearly independent. The integral of the function round any 

curve in the finite part of the plane, which does not 

include e l5 e< 2 or e 3 within its boundary, is zero, by Ex. 2; 

and this curve can be deformed to the shape in the figure, 

until it becomes infinitely large, without changing the 

value of the integral. 

Since the limit of zw for \z\ = 00 is zero, the value of 
the integral from oo to oo is zero, by 24, II. ; and if the 
description begin with a branch w, the branch at oo is -w. 
The rest of the integral consists of the sum of the values Fig. 29. 

round the loops, which is 




because a path round a loop changes the branch of w and the last branch after describing the 
loop round e 3 is +w at GO , the proper value ( 90, in). Hence, as the whole integral 
is zero, we have 



or say E 2 = 

Thus the value of the integral for any circuit Q, which restores the initial branch of w, can 
be expressed in any of the equivalent forms mE^ n E 3 , m E^n E^ m"E 2 + ri E z , where 
the m s and ris, are integers. 

Now any path from co to z can be resolved into a circuit fl, which restores at oo the 
initial branch of w, combined with either 

(i) a straight path from oo to 2, 

or (ii) a loop between oo and e 1} together with a straight path from oo to z. 
(The apparently distinct alternatives, of a loop between oo and e 2 together with a straight 
: path from oo to z and of a similar path round e a , are inclusible in the second alternative 
above ; the reasons are similar to those in Ex. 5.) 

f x 

If u denote j ^ {^(z-ej (z-e 2 ) (z-e 3 }}~*dz when the integral is taken in a straight 

, line, then the value of the integral for part (i) of a path is u; and the value of the 
1 integral for part (ii) of a path is E l - u, the initial branch in each case for these parts being 
. the initial branch of w for the whole path. Hence the most general value of the integral 
for any path is 

+ 2no> 3 + u, 



or 



the two being evidently included in the form 

2mo> 1 + 2n(,) 3 u. 
If, then, we denote by z = ft>(u) the relation which is inverse to 



we 

In the same way as in the preceding example, it follows that 



where ^ () is - {4 (z - e^ (z - e 2 ) (z - e 3 )}*. 



198 SIMPLE PERIODICITY [104 

The foregoing simple examples are sufficient illustrations of the multi 
plicity of value of an integral of a uniform function or of a multiform 
function, when branch-points or discontinuities occur in the part of the plane 
in which the path of integration lies. They also shew one of the modes in 
which singly-periodic and doubly-periodic functions arise, the periodicity 
consisting in the addition of arithmetical multiples of constant quantities 
to the argument. And it is to be noted that, as only a single value of z 
is used in the integration, so only a single value of z occurs in the 
inversion ; that is, the functions just obtained are uniform functions of their 
variables. To the properties of such periodic functions we shall return in the 
succeeding chapters. 

105. We proceed to the theory of uniform periodic functions, some 
special examples of which have just been considered ; and limitation will 
be made here to periodicity of the linear additive type, which is only a very 
special form of periodicity. 

A function f(z) is said to be periodic when there is a quantity &> such 
that the equation 

/(* + )=/(*) 

is an identity for all values of z. Then/0 + nw) =f(z), where n is any 
integer positive or negative; and it is assumed that &> is the smallest 
quantity for which the equation holds, that is, that no submultiple of &> will 
satisfy the equation. The quantity u> is called a period of the function. 

A function is said to be simply-periodic when there is only a single 
period : to be doubly-periodic when there are two periods ; and so on, the 
periodicity being for the present limited to additive modification of the 
argument. 

It is convenient to have a graphical representation of the periodicity of a 
function. 

(i) For simply-periodic functions, we 
take a series of points 0, A 1} A 2 ,..., 
A-i, ^4_ 2 ,... representing 0, w, 2o>, ... , 
<, 2&>, . . . ; and through these points 
we draw a series of parallel lines, dividing 
the plane into bands. Let P be any 
point z in the band between the lines 
through and through A^\ through P 
draw a line parallel to OA l and measure 




each equal to OA^ then all the points / 

P 1} P 2 , ... , P_i, P- 2 , ... are represented 

by z + nco for positive and negative integral values of n. But/ (2 + &)=/(*)] 

and therefore the value of the function at a point P n in any of the bands is 



105.] 



DOUBLE PERIODICITY 



199 



the same as the value at P. Moreover to a point in any of the bands there 
corresponds a point in any other of the bands ; and therefore, owing to the 
periodic resumption of the value at the points corresponding to each point P, 
it is sufficient to consider the variation of the function for points within one 
band, say the band between the lines through and through AI. A point P 
within the band is sometimes called irreducible, the corresponding points P 
in the other bands reducible. 

If it were convenient, the boundary lines of the bands could be taken 
through points other than A l} A 2 , ... ; for example, through points (m + |) &> 
for positive and negative integral values of ra. Moreover, they need not be 
straight lines. The essential feature of the graphic representation is the 
division of the plane into bands. 

(ii) For doubly-periodic functions a similar method is adopted. Let &> 
and co be the two periods of such a 
function /(#), so that 

/<.+)/(*)-/(+ <0; 

then f(z + nw + n w) =f(z), 
where n and n are any integers positive 
or negative. 

For graphic purposes, we take points 
0, A-L, A 2 , ..., A^i, A_ 2 , ... representing 
0, ft), 2&), . . . , to, 2(w, . . . ; and we take 
another series 0, B 1} B 2 , . . . , B_ 1} B_ 2 , . . . 
representing 0, &) , 2&/, . . . , ft/, 2ft/, . . . ; 
through the points A we draw lines 
parallel to the line of points B, and 
through the points B we draw lines 
parallel to the line of points A. The intersection of the lines through A n 
and B n > is evidently the point n&> + w &> , that is, the angular points of the 
parallelograms into which the plane is divided represent the points nco + n w 
for the values of n and n . 

Let P be any point z in the parallelogram OAfi-JS^ ; on lines through P, 
parallel to the sides of the parallelogram, take points Q 1} Q 2 , ... , Q_ lt Q_ 2 , ... 
such that PQ l = QiQ 2 = ... = ft) and points R lt R 2 , ... , R_ lt R_ 2 , ... such that 
PR l = R^ . . . to ; and through these new points draw lines parallel to 
the sides of the parallelogram. Then the variables of the points in which 
these lines intersect are all represented by z + mw + mV for positive and nega 
tive integral values of m and m ; and the point represented by z + m^ + m a) 
is situated in the parallelogram, the angular points of which are mw 4 mot , 
(m + 1) &) + mw, mco + (mf + 1) ft) , and (m -f 1) &) + (m + 1) ft/, exactly as P 
is situated in OA^C^. But 

/ (z + m^ + Wj V) = / (z\ 




Fig. 31. 



200 RATIO OF THE PERIODS [105. 

and therefore the value of the function at such a point is the same as the 
value at P. Since the parallelograms are all equal and similarly situated. 
to any point in any of them there corresponds a point in OA^G^B^; and the 
value of the function at the two points is the same. Hence it is sufficient to 
consider the variation of the function for points within one parallelogram, say, 
that which has 0, &>, o) + , &> for its angular points. A point P within 
this parallelogram is sometimes called irreducible, the corresponding points 
within the other parallelograms reducible to P ; the whole aggregate of the 
points thus reducible to any one are called homologous points. And the 
parallelogram to which the reduction is made is called the parallelogram of 
periods. 

As in the case of simply-periodic functions, it may prove convenient to 
choose the position of the fundamental parallelogram so that the origin is 
not on its boundary ; thus it might be the parallelogram the middle points of 
whose sides are + &>, + ^co . 

106. In the preceding representation it has been assumed that the line 
of points A is different in direction from the line of points B. If &> = u + iv 
and to = u +iv , this assumption implies that v /u is unequal to v/u, and 
therefore that the real part of a> /ia> does not vanish. The justification of 
this assumption is established by the proposition, due to Jacobi * : 

The ratio of the periods of a uniform doubly -periodic function cannot be 
real. 

Let/ (2) be a function, having CD and CD as its periods. If the ratio w /to 
be real, it must be either commensurable or incommensurable. 

If it be commensurable, let it be equal to n /n, where n and n are 
integers, neither of which is unity owing to the definition of the periods CD 
and 6Dj. 

Let n /n be developed as a continued fraction, and let m fm be the last 
convergent before n jn, where m and mf are integers. Then 

n m _ 1 

n m mn 
that is, mn - m n = 1, 

, 1 / . U> , . .. CD 

so that mco ~ mco = -(mn~ run ) = - . 

n x n 

Therefore f(z) =f(z + m co ~ mco ), 

since m and m are integers ; so that 

-, , ~( co\ 

/(*)-/( -i- s). 

contravening the definition of CD as a period, viz., that no submultiple of co is a 
period. Hence the ratio of the periods is not a commensurable real quantity. 

* Ges. Werke, t. ii, pp. 25, 26. 



106.] OF A UNIFORM DOUBLY-PERIODIC FUNCTION 201 

If it be incommensurable, we express oj /aj as a continued fraction. Let 
p/q and p /q be two consecutive convergents : their values are separated by 
the value of &> /&>, so that we may write 

v~q + \q ~q) 

where 1 > h > 0. 

Now pq <- p q 1, so that 



- = P + 
o> q qq 



where e is real and |e < 1 ; hence 



, e 

qa) pa) = , &>. 



Therefore f(z) =f(z + qw pa), 

since p and q are integers ; so that 



Now since &> /&> is incommensurable, the continued fraction is unending. We 
therefore take an advanced convergent, so that q is very large. Then - &> is 

a very small quantity and z + - &> is a point infinitesimally near to z, that 

is, the function / (V), under the present hypothesis, resumes its value at a 
point infinitesimally near to z. Passing along the line joining these two 
points infinitesimally near another, we should have / (z) constant along a 
line and therefore ( 37) constant everywhere ; it would thus cease to be a 
varying function. 

The ratio of the periods is thus not an incommensurable real quantity. 

We therefore infer Jacobi s theorem that the ratio of the periods cannot 
be real. In general, the ratio is a complex quantity ; it may, however, be a 
pure imaginary*. 

COROLLARY. If a uniform function have two periods w l and &> 2 such that 
a relation 

m l w l + ra 2 G> 2 = 

exists for integral values of m 1 and ?n 2 , the function is only simply-periodic. 
And such a relation cannot exist between two periods of a simply-periodic 
function, if m^ and ra 2 be real and incommensurable ; for then the function 
would be constant. 

* It was proved, in Ex. 5 and Ex. 6 of 104, that certain uniform functions are doubly-periodic. 
A direct proof, that the ratio of the distinct periods of the functions there obtained is not a real 
quantity, is given by Falk, Acta Math., t. vii, (1885), pp. 197200, and by Pringsheim, Math. 
Ann., t. xxvii, (1886), pp. 151157. 



202 UNIFORM [106. 

Similarly, if a uniform function have three periods &> 1; a>. 2> o> 3 , connected 

by two relations 

.. = 0, 



n 1 o) 1 + n 2 a) 2 + n 3 a) 3 = 0, 

where the coefficients m and n are integers, then the function is only simply- 
periodic. 

107. The two following propositions, also due to Jacobi*, are important 
in the theory of uniform periodic functions of a single variable : 

If a uniform function have three periods w^, 2 , MS such that a relation 

m^i + m. 2 &> 2 + m 3 w 3 = 

is satisfied for integral values ofm lt w 2 , m 3 , then the function is only a doubly- 
periodic function. 

What has to be proved, in order to establish this proposition, is that two 
periods exist of which w l , &> 2 , &> 3 are integral multiple combinations. 

Evidently we may assume that m^, ra 2 , m 3 have no common factor: let / 
be the common factor (if any) of m. 2 and m 3 , which is prime to m^. Then 
since 



and the right-hand side is an integral combination of periods, it follows that 
riod. 

is a fraction in its lowest terms. Change it into a continued 



-~ &>! is a period. 



fraction and let ^ be the last convergent before the proper value ; then 
2 



1 

so that <l~f~P = ^f- 

But o>! is a period and ^ft)! is a period; therefore q ^ Wj pwi is a period, 

or &>!// is a period, = to/ say. 

Let ra 2 //= m 2 , m 3 /f= m/, so that m 1& V + m 2 &> 2 + ??i 3 &) 3 = 0. Change 

fy 

m. 2 /m 3 into a continued fraction, taking - to be the last convergent before the 

proper value, so that 

m/ r _ 1 

/ i 



s sm s 

* Ges. Werke, t. ii, pp. 2732. 



107.] DOUBLY-PERIODIC FUNCTIONS 203 

Then r&> 2 + sco., being an integral combination of periods, is a period. But 
&> 2 = &) 2 (sm 2 r rm 3 ) 

= ra>. 2 m 3 s (m^ + w 3 &> 3 ) 

= m^sw-i - m a (r&> 2 + su> 3 ) ; 
also + ft) 3 = &) 3 (sm/ rm 3 ) 

sm 2 o) 3 + r (mjO)/ 



and o>! =/&)/. 

Hence two periods &>/ and r<u 2 + s&> 3 exist of which co 1} co 2 , &> 3 are integral 
multiple combinations ; and therefore all the periods are equivalent to &>/ and 
r&> 2 + so) 3 , that is, the function is only doubly-periodic. 

COROLLARY. If a function have four periods <u l3 &> 2 , co s , &> 4 connected by 
two relations 

m 1 o) 1 + m 2 o) 2 + wi 3 ft) 3 + ra 4 &> 4 = 0, 

72J60J + W 2 0) 2 + W 3 ft) 3 + W 4 4 = 0, 

where the coefficients m and w are integers, the function is only doubly- 
periodic. 

108. If a uniform function of one variable have three periods a) l , w.,, &> 3 , 
then a relation of the form 

m 1 o) l + w? 2 to 2 + in 3 (i) 3 = 
must be satisfied for some integral values ofm l} m 2 , m s . 

Let a) r = a r + i@ r , for r = 1, 2, 3 ; in consequence of 106, we shall assume 
that no one of the ratios of twj, <w 2 , w 3 in pairs is real, for, otherwise, either 
the three periods reduce to two immediately, or the function is a constant. 
Then, determining two quantities A, and fj, by the equations 



so that X and //, are real quantities and neither zero nor infinity, we have 



for real values of X and p. 

Then, first, if either X or fj. be commensurable, the other is also commen 
surable. Let X = a/6, where a and b are integers ; then 



= bo) 3 aa) } , 

so that fyu,&> 2 is a period. Now, if b/j, be not commensurable, change it into a 
continued fraction, and let p/q, p /q be two consecutive convcrgents, so that, 
as in 106, 

/ P , x 
bfji,=^+ ,, 

q qq 



204 TRIPLY-PERIODIC UNIFORM [108. 

where 1 > x > 1. Then - &>., + - ? is a period, and so is <w 2 ; hence 

q qq 

P~ ^ x 



IT 

is a period, that is, - <a 2 is a period. We may take q indefinitely large, and 

then the function has an infinitesimal quantity for a period, that is, it would 
be a constant under the hypothesis. Hence &/* (and therefore /*) cannot be 
incommensurable, if X be commensurable; and thus X and //. are simul 
taneously commensurable or simultaneously incommensurable. 

CL G 

If X and fj, be simultaneously commensurable, let X = j- , p = -^ , so that 

a c 

&) 3 = r &>! + -jG>2. 

o a 

and therefore 6rfto 3 = ac^ + bca) 2 , 

a relation of the kind required. 

If X and //. be simultaneously incommensurable, express A, as a continued 
fraction ; then by taking any convergent r/s, we have 

r _ x 
* = * 

/Yt 

where 1 > x > 1, so that s\ r=-: 

s 

by taking the convergent sufficiently advanced the right-hand side can be 
made infinitesimal. 

Let i\ be the nearest integer to the value of s/j,, so that, if 



we have A numerically less than ^. Then 

x 

sat-, ra> 1 r 1 w 2 = a) 1 + Aw.,, 
s 

fp 

and the quantity - Wj can be made so small as to be negligible. Hence 

S 

integers r, r lt s can be chosen so as to give a new period &>/(= A&> 2 ), such 
that | &)/ < \ &) 2 . 

We now take w l , &> 2 , &> 3 : they will be connected by a relation of the form 

0> 3 =X (W 1 +yLt / G) 2 / , 

and X and // must be incommensurable : for otherwise the substitution for 
to/ of its value just obtained would lead to a relation among a> l) &>o, &> 3 that 
would imply commensurability of X and of p. 

Proceeding just as before, we may similarly obtain a new period &> 2 " such 
that <o 2 " < \ ! m z I an d so on in succession. Hence we shall obtain, after n 



108.] FUNCTIONS DO NOT EXIST 205 

such processes, a period co 2 (w) such that |&) 2 (n) | < ^ a>*\, so that by making n 

z 

sufficiently large we shall ultimately obtain a period less than any assigned 
quantity. Let such period be to ; then 

/(*+)-/(*), 

and so for points along the co-line we have an infinite number close together 
at which the function is unaltered in value. The function, being uniform, 
must in that case be constant. 

It thus appears that, if A. and /j, be simultaneously incommensurable, the 
function is a constant. Hence the only tenable result is that A. and //. are 
simultaneously commensurable, and then there is a period-equation of the 

form 

m^w^ + m. 2 o) 2 + m 3 o) s = 0, 

where m 1 , w 2 , m 3 are integers. 

The foregoing proof is substantially due to Jacobi (I.e.). The result can 
be obtained from geometrical considerations by shewing that the infinite 
number of points, at which the function resumes its value, along a line 
through z parallel to the two-line will, unless the condition be satisfied, reduce 
to an infinite number of points in the a) 1 , &) 2 parallelogram which will form 
either a continuous line or a continuous area, in either of which cases the 
function would be a constant. But, if the condition be satisfied, then the 
points along the line through z reduce to only a finite number of points. 

COROLLARY I. Uniform functions of a single variable cannot have three 
independent periods ; in other words, triply -periodic uniform functions of a 
single variable do not exist* ; and, a fortiori, uniform functions of a single 
variable with a number of independent periods greater than two do not exist. 

But functions involving more than one variable can have more than two 
periods, e.g., Abelian transcendents ; and a function of one variable, having 
more than two periods, is not uniform. 

COROLLARY II. All the periods of a uniform periodic function of a single 
variable reduce either to integral multiples of one period or to linear combina 
tions of integral multiples of two periods whose ratio is not a real quantity. 

109. It is desirable to have the parallelogram, in which a doubly- 
periodic function is considered, as small as possible. If in the parallelogram 
(supposed, for convenience, to have the origin for an angular point) there be 
a point a)" such that 

/(* + ")=/(*) 
for all values of z, then the parallelogram can be replaced by another. 

* This theorem is also due to Jacobi, (I.e., p. 202, note). 



206 FUNDAMENTAL PARALLELOGRAM [109. 

It is evident that co" is a period of the function ; hence ( 108) we must 

have 

co" = Aco + /AW ; 

and both X and /JL, which are commensurable quantities, are less than unity 
since the point is within the parallelogram. Moreover, co -f co <", which 
is equal to (1 A,) co + (1 /")&> , is another point within the parallelogram; 
and 

/(* + + -")/(*), 

since co, co , co" are periods. Thus there cannot be a single such point, unless 

X = \ = p. 

But the number of such points within the parallelogram must be finite ; 
if there were an infinite number, they would form a continuous line or a 
continuous area where the uniform function had an unvarying value, and 
consequently ( 37) the function would have a constant value everywhere. 

To construct a new parallelogram when all the points are known, we first 
choose the series of points parallel to the co-line through the origin 0, and of 
that series we choose the point nearest 0, say A l . We similarly choose the 
point, nearest the origin, of the series of points parallel to the co-line and 
nearest to it after the series that includes A l} say B l : we take OA 1} OB 1 as 
adjacent sides of the parallelogram, and these lines as the vectorial repre 
sentations of the periods. No point lies within this parallelogram where the 
function has the same value as at ; hence the angular points of the original 
parallelograms coincide with angular points of the new parallelograms. 

When a parallelogram has thus been obtained, containing no internal 
point fl such that the function can satisfy the equation 



for all values of z, it is called a fundamental, or a primitive, parallelogram, : 
and the parallelogram of reference in subsequent investigations will be 
assumed to be of a fundamental character. 

But a fundamental parallelogram is not unique. 

Let co and co be the periods for a given fundamental parallelogram, so 
that every other period co" is of the form Aco + //-co , where A, and /* are 
integers. Take any four integers a, b, c, d such that ad lc=l, as may 
be done in an infinite variety of ways ; and adopt two new periods coj and co 2 , 
such that 

&>! = aco + bo) , co 2 = ceo + d(o . 

Then the parallelogram with coj and co 2 for adjacent sides is fundamental. 
For we have 

+ eo = do) 1 ba> 2 , + co = cco x + aco 2 , 

and therefore any period co" 
= A.CO + /uco 
= (\d - fie) w l + ( \b + fj.a) eo 2 , save as to signs of A, and /z. 



109.] OF PERIODS 207 

The coefficients of o^ and &) 2 are integers, that is, the point <w" lies outside 
the new parallelogram of reference; there is therefore no point in it such that 

/(* + *>")=/(*), 
and hence the parallelogram is fundamental. 

COROLLARY. The aggregate of the angular points in one division of the 
plane into fundamental parallelograms coincides with their aggregate in 
any other division into fundamental parallelograms ; and all fundamental 
parallelograms for a given function are of the same area. 

The method suggested above for the construction of a fundamental parallelogram is 
geometrical, and it assumes a knowledge of all the points w" within a given parallelogram 
for which the equation/ (z -f ")=/ (z) is satisfied. 

Such a point o> 3 within the o^, o> 2 parallelogram is given by 

nil m 2 

<Bo= - (Bi -\ -- 0> 9 , 

m 3 J m 3 2 

where & 1} m 2 , m 3 are integers. We may assume that no two of these three integers 
have a common factor; were it otherwise, say for m^ and wi 2 , then, as in 107, a 
submultiple of o> 3 would be a period a result which may be considered as excluded. 
Evidently all the points in the parallelogram are the reduced points homologous with 
w 3 , 2o> 3 , ...... , (m 3 1) 3 ; when these are obtained, the geometrical construction is 

possible. 

The following is a simple and practicable analytical method for the construction. 

Change w^/rag and m z /m 3 into continued fractions; and let p/q and r/s be the 
last convergents before the respective proper values, so that 

m x p e m 2 r f 

m 3 q gm 3 m 3 s sm 3 
where e and e are each of them +1. Let 

m "> n , M m l j , ^ 

q =d + , s^ = $+ , 

m 3 m 3 m 3 m 3 

where X and p, are taken to be less than m 3 , but they do not vanish because q and s are 
less than m 3 . Then 

2 eo 3 -^w 1 -(9o)2 = (/*a> 2 + fi), *a>3-ro> 2 -<<>! = (Xa^ + e tOjj) ; 

U vn II io 

the left-hand sides are periods, say Q x and O 2 respectively, and since /u + e is not >m 3 and 
X + e is not >m 3 , the points Q. l and Q 2 determine a parallelogram smaller than the initial 
parallelogram. 



Thus 
are equations defining new periods Q ly Q 2 . Moreover 

, . X m-. p 65 a m 9 r t o 

4>-\ -- = s-^=s*-+ -, + -f^ = n ? = - + -L : 

m 3 m 3 q qm s m 3 2 m s * s sm 3 

so that, multiplying the right-hand sides together and likewise the left-hand sides, we 
at once see that X/i-ee is divisible by m s if it be not zero: let 

X/i ee = wi 3 A. 

Then, as X and p are less than m 3 , they are greater than A; and they are prime to it, 
because ee is +1. 



208 MULTIPLE [109. 

Hence we have Aa>j = ^Q 2 - t Q l , Aa> 2 = XQ 1 - ei V 

Since X and /u are both greater than A, let 

X = X 1 A + X , /x = /i 1 A + //, 
where X and /x are <A. Then X /* i g divisible by A if it be not zero, say 

X p - ee = AA ; 
then X and p. are >A and are prime to it. And now 

A (wj /^iO 2 ) = /x Q 2 ~ e ^i > A ( W 2 "~ ^1^1) = ^- QI ~ f 2 i 

and therefore, if (a 1 /^G^Qg, <B 2 -X 1 Q 1 = Q 4 , which are periods, we have 



With Q 3 and Q 4 we can construct a parallelogram smaller than that constructed 
with Qj and Q 2 . 

We now have A Q 1 = fG 3 +//G 4 , A Q.j=X Q 3 + e fl 4 , 

that is, equations of the same form as before. We proceed thus in successive stages : 
each quantity A thus obtained is distinctly less than the preceding A, and so finally we 
shall reach a stage when the succeeding A would be unity, that is, the solution of the pair 
of equations then leads to periods that determine a fundamental parallelogram. It 
is not difficult to prove that a> lt o> 2 , o> 3 are combinations of integral multiples of these 
periods. 

If one of the quantities, such as X /x -ee , be zero, then X =/x = l, e = e = 1 ; and then 
Q 3 and O 4 are identical. If e = e = + 1, then AQ 3 = Q 2 - Qj , and the fundamental parallelo 
gram is determined by 

<V = QI + - (Q 2 - %), G 4 = Q 2 - 1 (Q 2 - Q t )- 

If f = f = -1, then AQ 3 = Q 2 +O 15 so that, as A is not unity in this case, the fundamental 
parallelogram is determined by Q 2 and Q 3 . 

Ex. If a function be periodic in a> 1? a> 2 , and also in <o 3 where 

29co = 1 7 



periods for a fundamental parallelogram are 

QI = Scoj + 3o) 2 - 8w 3 , Q 2 = 3 eoj + 2co 2 - 5w 3 , 
and the values of a> 1} < 2 , w 3 in terms of O/ and Q 2 are 



G) 2 = 2 - 1 , a> 3 = 2 Q. 

Further discussion relating to the transformation of periods and of fundamental 
parallelograms will be found in Briot and Bouquet s The orie des fonctions elliptujues, 
pp. 234, 235, 268272. 

110. It has been proved that uniform periodic functions of a single 
variable cannot have more than two periods, independent in the sense that 
their ratio is not a real quantity. If then a function exist, which has two 
periods with a real incommensurable ratio or has more than two independent 
periods, either it is not uniform or it is a function (whether uniform or multi 
form) of more variables than one. 

When restriction is made to uniform functions, the only alternative is 
that the function should depend on more than one variable. 



110.] PERIODICITY 209 

In the case when three periods o) l , &> 2 , &> 3 (each of the form a-f t/3) were 
assigned, it was proved that the necessary condition for the existence of a 
uniform function of a single variable is that finite integers m ly m 2 , m :i can 
be found such that 

ra 2 cr 2 + m 3 a 3 = 0, 

- w 3 /3 3 = ; 

and that, if these conditions be not satisfied, then finite integers m 1} m. 2 , m s 
can be found such that both Sma and 2m/8 become infinitesimally small. 

This theorem is purely algebraical, and is only a special case of a more 
general theorem as follows : 

Let a n , or 12 ,..., a lt r+1 ; a 21 , a w ,..., a 2>r+1 ;...; a rl , .,..., a r ,r+i be r sets of 
real quantities such that a relation of the form 

w i a *i + ^2^2 + . + n r+l <x sr+ i = 

is not satisfied among any one set. Then finite integers m^,..., m r+1 can be 
determined such that each of the sums 

j (for 5 = 1, 2,...,r) is an infinitesimally small quantity. And, a fortiori, if 
fewer than r sets, each containing r+1 quantities be given, the r+1 
integers can be determined so as to lead to the result enunciated ; all that 

i is necessary for the purpose being an arbitrary assignment of sets of real 

| quantities necessary to make the number of sets equal to r. But the result 

! is not true if more than r sets be given. 

We shall not give a proof of this general theorem* ; it would follow the 
lines of the proof in the limited case, as given in 108. But the theorem 
will be used to indicate how the value of an integral with more than 
two periods is affected by the periodicity. 

Let / be the value of the integral taken along some assigned path from 
an initial point Z Q to a final point z\ and let the periods be &) 1} &> 2 ,..., &> r , 
(where r > 2), so that the general value is 

/ + fftjcoj + m 2 a)., + . . . + m r w r , 

where m lt m 2 ,..., m r are integers. Now if co s = a s + i/3 s , for s=l, 2,..., r, 
when it is divided into its real and its imaginary parts, then finite integers 
Wi, n 2 ,..., n r can be determined such that 



and n-if. 

are both infinitesimal ; and then 



2 n s 



is infinitesimal. But the addition 



of S n s co s still gives a value of the integral ; hence the value can be modified 

* A proof will be found in Clebsch and Gordan s Theorie der Abel schen Functioncn, 38. 
F - 14 



210 MULTIPLE PERIODICITY [110. 

by infinitesimal quantities, and the modification can be repeated indefinitely. 
The modifications of the value correspond to modifications of the path from 
Z Q to z ; and hence the integral, regarded as depending on a single variable, 
can be made, by modifications of the path of the variable, to assume any 
value. The integral, in fact, has not a definite value dependent solely 
upon the final value of the variable; to make the value definite, the path 
by which the variable passes from the lower to the upper limit must be 
specified. 

It will subsequently ( 239) be shewn how this limitation is avoided by 
making the integral, regarded as a function, depend upon a proper number 
of independent variables the number being greater than unity. 

Ex. 1. If F be the value of i -, , (n integral), taken along an assigned path, 

Jo (\-z n Y 
and if 

P = 2 I 1 - ^-j(# real), 

then the general value of the integral is 

I \ ^ I 

n 

where q is any integer and m p any positive or negative integer such that 2 m p = 0. 

P=I 

(Math. Trip. Part II, 1889.) 

Ex. 2. Prove that v= I udz, where 
J o 

is an algebraical function satisfying the equation 

and obtain the conditions necessary and sufficient to ensure that 

i) = fadz 
should be an algebraical function, when u is an algebraical function satisfying an equation 

(Liouville, Briot and Bouquet.) 



CHAPTER X. 



SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS. 



111. ONLY a few of the properties of simply-periodic functions will be 
given, partly because some of them are connected with Fourier s series the 
detailed discussion of which lies beyond our limits, and partly because, as 
will shortly be explained, many of them can at once be changed into 
properties of uniform non-periodic functions which have already been 
considered. 

When we use the graphical method of 105, it is evident that we need 
consider the variation of the function within only a single band. Within 
that band any function must have at least one infinity, for, if it had not, it 
would not have an infinity anywhere in the plane and so would be a constant ; 
and it must have at least one zero, for, if it had not, its reciprocal, also a 
simply-periodic function, would not have an infinity in the band. The 
infinities may, of course, be accidental or essential : their character is repro 
duced at the homologous points in all the bands. 

For purposes of analytical representation, it is convenient to use a 
relation 

Ziri 

so that, if the point Z in its plane have R and (*) 
for polar coordinates, 



, 

Z = = ; log R + 

Z7TI 



ft). 



If we take any point A in the ^-plane and a 
corresponding point a in the z-plane, then, as Z 
describes a complete circle through A with the 
origin as centre, z moves along a line aa l} where 
di is a + a). A second description of the circle 
makes z move from a x to a a , where a 2 = a x + &> 




Fig. 32. 

and so on in succession. 
142 



212 SIMPLE PERIODICITY [111. 

For various descriptions, positive and negative, the point a describes a line, 
the inclination of which to the axis of real quantities is the argument of &>. 

Instead of making Z describe a circle through A, let us make it describe 
a part of the straight line from the origin through A, say from A, where 
OA = R, to C, where 00 = R . Then z describes a line through a perpend 
icular to aa l} and it moves to c where 



Similarly, if any point A on the former circumference move radially to a 
point C at a distance R from the ^-origin, the corresponding z point a 
moves through a distance a c , parallel and equal to ac : and all the points c 
lie on a line parallel to aa^. Repeated description of a ^-circumference with 
the origin as centre makes z describe the whole line cCjCo. 

If then a function be simply-periodic in &>, we may conveniently take 
any point a, and another point a^ = a + w, through a and a^ draw straight 
lines perpendicular to aa 1} and then consider the function within this band. 
The aggregate of points within this band is obtained by taking 

(i) all points along a straight line, perpendicular to a boundary of 

the band, as aa^ ; 
(ii) the points along all straight lines, which are drawn through the 

points of (i) parallel to a boundary of the band. 

In (i), the value of z varies from to co in an expression a + z, that is, in 
the ^-plane for a given value of R, the angle varies from to 2?r. 

In (ii), the value of log R varies from oo to +00 in an expression 
fi\ 

. log R + = w, that is, the radius R must vary from to oo . 

2?r 



Hence the band in the 0-plane and the whole of the ^-plane are made 
equivalent to one another by the transformation 



Now let z be any special point in the finite part of the band for a given 
simply-periodic function, and let Z be the corresponding point in the Z-planej 
Then for points z in the immediate vicinity of z and for points Z which 
are consequently in the immediate vicinity of Z , we have 



Ziri 



e 
to 



where | X differs from unity only by an infinitesimal quantity. 



111.] FOURIER S THEOREM 213 

If then w, a function of z, be changed into W a function of Z, the following 
relations subsist : 

When a point Z Q is a zero of w, the corresponding point Z Q is a zero 

of W. 

When a point z is an accidental singularity of w, the corresponding 
point Z is an accidental singularity of W. 

When a point z is an essential singularity of w, the corresponding 
point Z is an essential singularity of W. 

When a point z is a branch- point of any order for a function w, the 
corresponding point Z is a branch-point of the same order for W. 

And the converses of these relations also hold. 

Since the character of any finite critical point for w is thus unchanged by the 
transformation, it is often convenient to change the variable to Z so as to let 
the variable range over the whole plane, in which case the theorems already 
proved in the preceding chapters are applicable. But special account must 
be taken of the point z = oo . 

112. We can now apply Laurent s theorem to deduce what is practically 

Fourier s series, as follows. 

Let f(z) be a simply-periodic function having w as its period, and suppose 
that in a portion of the z-plane bounded by any two parallel lines, the inclina 
tion of which to the axis of real quantities is equal to the argument of w, the 
function is uniform and has no singularities; then, at points within that 
portion of the plane, the function can be expressed in the form of a converging 

2n-2t 

series of positive and of negative integral powers of e " . 

In figure 32, let aa^a^... and cc^... be the two lines which bound the 
portion of the plane : the variations of the function will all take place within 
that part of the portion of the plane which lies within one of the repre 
sentative bands, say within the band bounded by ...ac... and . ..a^...: that is, 
we may consider the function within the rectangle acc^a, where it has no 
singularities and is uniform. 

Now the rectangle acc^a in the 2-plane corresponds to a portion of the 
Z-plane which, after the preceding explanation, is bounded by two circles 

2iri 2irf 

with the origin for common centre and of radii | e w " | and | e u ; and the 
variations of the function within the rectangle are given by the variations of 
a transformed function within the circular ring. The characteristics of the 
one function at points in the rectangle are the same as the characteristics of 
the other at points in the circular ring : and therefore, from the character 
of the assigned function, the transformed function has no singularities and it 



214 FOURIER S THEOREM [112. 

is uniform within the circular ring. Hence, by Laurent s Theorem ( 28), 
the transformed function is expressible in the form 



a series which converges within the ring : and the value of the coefficient a n 

is given by 

1 



tvfj Z n +* 

taken along any circle in the ring concentric with the boundaries. 

Retransforming to the variable z, the expression for the original function 
is 

71 = + oo Zrnriz 

f(z) = 2 a n e~^~ . 

71= -00 

The series converges for points within the rectangle and therefore, as it 
is periodic, it converges within the portion of the plane assigned. And the 
value of a n is 

Zniriz 
(?\P *~ d? 

\ z ) 6 az, 



taken along a path which is the equivalent of any circle in the ring concentric 
with the boundaries, that is, along any line a c perpendicular to the lines 
which bound the assigned portion of the plane. 

The expression of the function can evidently be changed into the form 

Znvi, _ 



1 r 

- 

Wj 7 



where the integral is taken along the piece of a line, perpendicular to the 
boundaries and intercepted between them. 

If one of the boundaries of the portion of the plane be at infinity, (so that 
the periodic function has no singularities within one part of the plane), then 
the corresponding portion of the ^-plane is either the part within or the part 
without a circle, centre the origin, according as the one or the other of the 
boundaries is at oo . In the former case, the terms with negative indices 
n are absent ; in the latter, the terms with positive indices are absent. 

113. On account of the consequences of the relation subsisting between 
the variables z and Z, many of the propositions relating to general uniform 
functions, as well as of those relating to multiform functions, can be changed, 
merely by the transformation of the variables, into propositions relating to 
simply-periodic functions. One such proposition occurs in the preceding 
section ; the following are a few others, the full development being unnecess 
ary here, in consequence of the foregoing remark. * The band of reference 
for the simply-periodic functions considered will be supposed to include the 



113.] SIMPLY-PERIODIC FUNCTIONS 215 

origin : and, when any point is spoken of, it is that one of the series of 
homologous points in the plane, which lies in the band. 

We know that, if a uniform function of Z have no essential singularity, 
then it is a rational algebraical function, which is integral if z = cc be the 
only accidental singularity and is meromorphic if there be accidental singu 
larities in the finite part of the plane ; and every such function has as many 
zeros as it has accidental singularities. 

Hence a uniform simply-periodic function with z=cc as its sole essential 
singularity has as many zeros as it has infinities in each band of the plane ; 
the number of points at which it assumes a given value is equal to the number 
of its zeros ; and, if the period be w, the function is a rational algebraical 

ZTTIZ 

function of e a , which is integral if all the singularities be at an infinite 
distance and is meromorphic if some (or all) of them be in a finite part of 
the plane. But any number of the zeros and any number of the infinities 
may be absorbed in the essential singularity at z = oo . 

The simplest function of Z, thus restricted to have the same number of 
zeros as of infinities, is one which has a single zero and a single infinity in 
the finite part of the plane ; the possession of a single zero and a single infinity 
will therefore characterise the most elementary simply-periodic function. 
Now, bearing in mind the relation 

Zniz 

Z=e<*, 

the simplest -pomt to choose for a zero is the origin, so that Z = 1 ; and then 
the simplest ^-point to choose for an infinity at a finite distance is \w, (being 
half the period), so that Z \. The expression of the function in the 
Z-plane with 1 for a zero and 1 for an accidental singularity is 

Z ~ l 



and therefore assuming as the most elementary simply-periodic function that 
which in the plane has a series of zeros and a series of accidental singularities 
all of the first order, the points of the one being midway between those of the 
other, its expression is 



A 



2iriz 

e" -I 



Zniz 

which is a constant multiple of tan . Since e " is a rational fractional 

CD 

function of tan , part of the foregoing theorem can be re-stated as follows: 

If the period of the function be o>, the function is a rational algebraical 

function of tan . 

n 



216 SIMPLY-PERIODIC [113. 

Moreover, in the general theory of uniform functions, it was found con 
venient to have a simple element for the construction of products, there 
( 53) called a primary factor: it was of the type 



^Z-u 

where the function G ( -~ j could be a constant; and it had only one infinity 

and one zero. 

Hence for simply-periodic functions we may regard tan as a typical 

primary factor when the number of irreducible zeros and the (equal) number 
of irreducible accidental singularities are finite. If these numbers should 
tend to an infinite limit, then an exponential factor might have to be 

associated with tan ; and the function in that case might have essential 
singularities elsewhere than at z = oo . 

114. We can now prove that every uniform function, which has no 
essential singularities in the finite part of the plane and is such that all its 
accidental singularities and its zeros are arranged in groups equal and 
finite in number at equal distances along directions parallel to a given 
direction, is a simply-periodic function. 

Let to be the common period of the groups of zeros and of singularities : 
and let the plane be divided into bands by parallel lines, perpendicular to 
any line representing w. Let a, b, ... be the zeros, a, /3, ... the singularities 
in any one band. 

Take a uniform function </> (z), simply-periodic in <w and having a single 
zero and a single singularity in the band : we might take tan as a value 
of <f> (z). Then 



is a simply-periodic function having only a single zero, viz., z = a and a single 
singularity, viz., z a. ; for as <f> {z} has only a single zero, there is only a single 
point for which (f>(z) = <f) (a), and a single point for which < (z) $ (a). Hence 



is a simply-periodic function with all the zeros and with all the infinities of 
the given function within the band. But on account of its periodicity it has 
all the zeros and all the infinities of the given function over the whole plane ; 
hence its quotient by the given function has no zero and no singularity over 
the whole plane and therefore it is a constant ; that is, the given function, 



114.] FUNCTIONS 217 

save as to a constant factor, can be expressed in the foregoing form. It is 
thus a simply-periodic function. 

This method can evidently be used to construct simply-periodic functions, having 
assigned zeros and assigned singularities. Thus if a function have a + mat as its zeros and 
c+m <o as its singularities, where m and m have all integral values from oo to +00, 
the simplest form is obtained by taking a constant multiple of 

TTZ 7T 

tan tan 



TTZ , TTC 
tan tan 



Ex. Construct a function, simply-periodic in w, having zeros given by (m+^)o> and 

)o> and singularities by (m + i)co and (m + ) co. 
The irreducible zeros are ^co and f w ; the irreducible singularities are \u> and . Now 



f.TTZ \ ( TTZ , \ 

I tan tan ATT I I tan tan |TT I 

/ \ <" / \ M / 

7TZ \ ( TTZ \ 

tan tan JTT ] ( tan tan |TT I 

/ \ / 



is evidently a function, initially satisfying the required conditions. But, as tari^r is 
infinite, we divide out by it and absorb it into A as a factor ; the function then takes 
the form 

1 + tan - 



3-tan 7 ^ 

60 

We shall not consider simply-periodic functions, which have essential 
singularities elsewhere than at z = <x> ; adequate investigation will be found 
in the second part of Guichard s memoir, (I.e., p. 147). But before leaving the 
consideration of the present class of functions, one remark may be made. It 
was proved, in our earlier investigations, that uniform functions can be 
expressed as infinite series of functions of the variable and also as infinite 
products of functions of the variable. This general result is true when the 
functions in the series and in the products are simply-periodic in the same 
period. But the function, so represented, though periodic in that common 
period, may also have another period : and, in fact, many doubly-periodic 
functions of different kinds ( 136) are often conveniently expressed as infinite 
converging series or infinite converging products of simply-periodic functions. 

Any detailed illustration of this remark belongs to the theory of elliptic functions : one 
simple example must suffice. 

, ima 

Let the real part of - - be negative, and let q denote e " ; then the function 



being an infinite converging series of powers of the simply-periodic function e " , is finite 
everywhere in the plane. Evidently 6 (z) is periodic in o>, so that 

= 6 (z). 



218 DOUBLY-PERIODIC [114. 







Again, 0(s + ) = 2 






the change in the summation so as to give $ (z) being permissible because the extreme 
terms for the infinite values of n can be neglected on account of the assumption with 
regard to q. There is thus a pseudo-periodicity for 6(z) in a period <> . 






Similarly, if s (z)= q* e 



2J7TZ 

6 3 (z + <a ) = -e " 6(z). 

Then 6 3 (z) -r-d(z) is doubly-periodic in w and 2co , though constructed only from 
functions simply-periodic in w : it is a function with an infinite number of irreducible 
accidental singularities in a band. 

115. We now pass to doubly-periodic functions of a single variable, the 
periodicity being additive. The properties, characteristic of this important 
class of functions, will be given in the form either of new theorems or 
appropriate modifications of theorems, already established ; and the develop 
ment adopted will follow, in a general manner, the theory given by Liouville*. 
It will be assumed that the functions are uniform, unless multiformity be 
explicitly stated, and that all the singularities in the finite part of the plane 
are accidental "f*. 

The geometrical representation of double-periodicity, explained in 105, 
will be used concurrently with the analysis; and the parallelogram of 
periods, to which the variable argument of the function is referred, is a 
fundamental parallelogram ( 109) with periods J 2co and 2&> . An angular 
point for the parallelogram of reference can be chosen so that neither a 
zero nor a pole of the function lies on the perimeter; for the number 
of zeros and the number of poles in any finite area must be finite, 
otherwise they would form a continuous line or a continuous area, or thej 
would be in the vicinity of an essential singularity. This choice will, ir 

* In his lectures of 1847, edited by Borchardt and published in Crelle, t. Ixxxviii, (1880), pp. 
277 310. They are the basis of the researches of Briot and Bouquet, the most complet 
exposition of which will be found in their Theorie des fonctions elliptiques, (2nd ed.), pp. 
239280. 

t For doubly-periodic functions, which have essential singularities, reference should be made 
to Guichard s memoir, (the introductory remarks aud the third part), already quoted on p. 147, note. 

J The factor 2 is introduced merely for the sake of convenience. 



115.] FUNCTIONS 219 

general, be made ; but, in particular cases, it is convenient to have the origin 
as an angular point of the parallelogram and then it not infrequently occurs 
that a zero or a pole lies on a side or at a corner. If such a point lie on a side, 
the homologous point on the opposite side is assigned to the parallelogram 
which has that opposite side as homologous; and if it be at an angular point, 
the remaining angular points are assigned to the parallelograms which have 
them as homologous corners. 

The parallelogram of reference will therefore, in general, have z , z + 2&>, 
z + 2&/, z + 2&> + 2&> for its angular points ; but occasionally it is desirable 
to .take an equivalent parallelogram having z &> + &> as its angular 
points. 

When the function is denoted by </> (2), the equations indicating the 
periodicity are 

< (z + 2<w) = (f> (z) = (f> (z + 2&/). 

116. We now proceed to the fundamental propositions relating to 
doubly-periodic functions. 

I. Every doubly -periodic function must have zeros and infinities within 
the fundamental parallelogram. 

For the function, not being a constant, has zeros somewhere in the plane 
and it has infinities somewhere in the plane ; and, being doubly-periodic, it 
experiences within the parallelogram all the variations that it can have over 
the plane. 

COROLLARY. The function cannot be a rational integral function of z. 

For within a parallelogram of finite dimensions an integral function has 
no infinities and therefore cannot represent a doubly-periodic function. 

An analytical form for <j) (z) can be obtained which will put its singu 
larities in evidence. Let a be such a pole, of multiplicity n ; then we know 
that, as the function is uniform, coefficients A can be determined so that the 
function 

f(* ~ (z-a) n ~ (z-a) n - 1 ~ "~(z-a) 2 ~ z^a 

is finite in the vicinity of a ; but the remaining poles of <j> (z) are singularities 
of this modified function. Proceeding similarly with the other singularities 
b, c,..., which are finite in number and each of which is finite in degree, we 
have coefficients A, B, C,... determined so that 

A^< V i? K 

9 (z) 2, f Z T r 

is finite in the vicinity of every pole of <f) (z) within the parallelogram and 
therefore is finite everywhere within the parallelogram. Let its value be 



220 PROPERTIES [116. 

%(X); then for points lying within the parallelogram, the function <f>(z) is 
expressed in the form 



+ A * + 


^ 1 1 


A, 


T T ; 

2 a ( 

ft 

+ 1 i 


\9 1 / 

z - a> ( 
B 2 


z - a) n 
B m 


X. / 

6 ( 


7 \ 1 / 


z-b) m 


H, 

_L _L 


# 2 


Si 



z-h^ (z-h? r (z-h) 1 

But though < (^) is periodic, ^ (2?) is not periodic. It has the property of 
being finite everywhere within the parallelogram ; if it were periodic, it 
would be finite everywhere, and therefore could have only a constant value ; 
and then <f> (z) would be an algebraical meromorphic function, which is not 
periodic. The sum of the fractions in $ (z) may be called the fractional 
part of the function : owing to the meromorphic character of the function, 
it cannot be evanescent. 

The analytical expression can be put in the form 
(z - a)~ n (z - 6)-. . .(z - h)~ l F(z\ 

where F(z) is finite everywhere within the parallelogram. If a, /3, ..., ij be 
all the zeros, of degrees v, p, ..., X, within the parallelogram, then 

F(z) = (z-a) v (z-py ...(z-^G(z\ 

where G (z) has no zero within the parallelogram ; and so the function can 
be expressed in the form 

(z-a) n (z-b} m ...(z-h) 1 G ^ 

where G (z) has no zero and no infinity within the parallelogram or on its 
boundary ; and G {z) is not periodic. 

The order of a doubly-periodic function is the sum of the multiplicities 
of all the poles which the function has within a fundamental parallelogram; 
and, the sum being n, the function is said to be of the nth order. All 
these singularities are, as already remarked, accidental; it is convenient 
to speak of any particular singularity as simple, double, . . . according to its 
multiplicity. 

If two doubly-periodic functions u and v be such that an equation 



is satisfied for constant values of A, B, C, the functions are said to be 
equivalent to one another. Equivalent functions evidently have the same 
accidental singularities in the same multiplicity. 

II. The integral of a doubly-periodic function round the boundary of a 
fundamental parallelogram is zero. 



116.] 



OF DOUBLY-PERIODIC FUNCTIONS 



221 



Let ABCD be a fundamental parallelogram, the boundary of it being 
taken so as to pass through no pole of the 
function. Let A be z , B be z +2ca, and* < = 

D be z + 2a) : then any point in AB is / 

/Q* Q, 



where is a real quantity lying between and 1 ; 
and therefore the integral along AB is 

rl 



Any point in EG is z + 2<w + 2&> , where is a real quantity lying between 
and 1 ; therefore the integral along BC is 



(o dt, 

o 

since <^> is periodic in 2&). 

Any point in DC is s + 2o> + 2<wZ, where < is a real quantity lying 
between and 1 ; therefore the integral along CD is 



f 

J 1 



2ft) 



= - I 

J o 

Similarly, the integral along DA is 

= - I cf> Oo + 2o> ) 2w ^. 

J o 

Hence the complete value of the integral, taken round the parallelogram, i 

fi 
= <j>(z 

Jo 



which ^ is manifestly zero, since each of the integrals is the integral of 
a continuous function. 

COROLLARY. Let ty(z) be any uniform function of z t not necessarily 
doubly-periodic, but without singularities on the boundary. Then the 

* The figure implies that the argument of w is greater than the argument of w, a 
hypothesis which, though unimportant for the present proposition, must be taken account of 
hereafter (e.g., 129). 



222 INTEGRAL RESIDUE [116. 



integral jty (z) dz taken round the parallelogram of periods is easily seen 
to be 



n ri 

^ (z (} + Scot) 2udt + I ^(z + 2a> + 2m t) 2a> dt 

Jo J o 

ri ri 

- V (*o + 2o> + 2a>t) 2(odt - ^ (z + 2to t) 2w dt ; 

Jo Jo 



or, if we write 



/ ri ri 

then U- (2) ^ = I I/TJ (> + 2w t) 2m dt - ^ (z, + 2wt) 2(odt, 

J Jo Jo 

where on the left-hand side the integral is taken positively round the 
boundary of the parallelogram and on the right-hand side the variable t 
in the integrals is real. 

The result may also be written in the form 

r rD rx 

\-^r(z)dz=\ ^ (z) dz I -K (z) dz, 

J J A J A 

the integrals on the right-hand side being taken along the straight lines AD 
and AB respectively. 

Evidently the foregoing main proposition is established, when -^ () and 
T/r 2 (f) vanish for all values of . 

III. If a doubly -periodic function $(z) have infinities Oj, a 2 , ... within 
the parallelogram, and if A l , A 2 , ... be the coefficients of (z e^)" 1 , (z a^r 1 , . . . 
respectively in the fractional part of (j> (z) when it is expanded in the parallelo 
gram, then 

A 1 + A 2 +...=0. 

As the function <f>(z) is uniform, the integral f(f>(z)dz is, by ( 19, II.), the 
sum of the integrals round a number of curves each including one and only 
one of the infinities within that parallelogram. 

Taking the expression for (f>(z) on p. 220, the integral A m f(z a)~ m dz 
round the curve enclosing a is 0, if m be not unity, and is Z>jriA l , if m be 
unity; the integral K m f(z k)~ m dz round that curve is for all values of m 
and for all points k other than a ; and the integral /^ (z) dz round the curve 
is zero, since % (z) is uniform and finite everywhere in the vicinity of a. Hence 
the integral of < (z) round a curve enclosing c^ alone of all the infinities is 



Similarly the integral round a curve enclosing a. 2 alone is 27riA. 2 ; and so 
on, for each of the curves in succession. 

Hence the value of the integral round the parallelogram is 

2-rnZA. 



116.] OF FUNCTIONS OF THE SECOND ORDER 223 

But by the preceding proposition, the value of /(/> (2) dz round the parallelo 
gram is zero ; and therefore 



This result can be expressed in the form that the sum of the residues* of a 
doubly -periodic function relative to a fundamental parallelogram of periods 
is zero. 

COROLLARY 1. A doubly-periodic function of the first order does not 

exist. 

Let such a function have a for its single simple infinity. Then an 
expression for the function within the parallelogram is 

A 

^-a + *^> 

where ^ (2) is everywhere finite in the parallelogram. By the above propo 
sition, A vanishes ; and so the function has no infinity in the parallelogram. 
It therefore has no infinity anywhere in the plane, and so is merely a 
constant : that is, qua function of a variable, it does not exist. 

COROLLARY 2. Doubly-periodic functions of the second order are of two 
classes. 

As the function is of the second order, the sum of the degrees of the 
infinities is two. There may thus be either a single infinity of the second 
degree or two simple infinities. 

In the former case, the analytical expression of the function is 



where a is the infinity of the second degree and ^ (z) is holomorphic within 
the parallelogram. But, by the preceding proposition, A 1 = 0; hence the 
analytical expression for a doubly-periodic function with a single irreducible 
infinity a of the second degree is 



(z - of T * v 

within the parallelogram. Such functions of the second order, which have 
only a single irreducible infinity, may be called the first class. 
In the latter case, the analytical expression of the function is 



where c, and c 2 are the two simple infinities and x( z } i g finite within the 
parallelogram. Then 



See p. 42. 



224 PROPERTIES OF FUNCTIONS [116. 

so that, if C l = - C. 2 = C, the analytical expression for a doubly-periodic 
function with two simple irreducible infinities a 1 and 2 i g 



n 
G 



( 1 1 

( - 

\z-a-L z - 



within the parallelogram. Such functions of the second order, which have 
two irreducible infinities, may be called the second class. 

COROLLARY 3. If within any parallelogram of periods a function is 
only of the second order, the parallelogram is fundamental. 

COROLLARY 4. A similar division of doubly -periodic functions of any 
order into classes can be effected according to the variety in the constitution of 
the order, the number of classes being the number of partitions of the order. 

The simplest class of functions of the nth order is that in which the 
functions have only a single irreducible infinity of the nth degree. Evi 
dently the analytical expression of the function within the parallelogram is 

G, G, G n 

(z - a) 2 (z - a) 3 (z - a) n * ^ 

where ^ (z) is holomorphic within the parallelogram. Some of the coefficients 
G may vanish ; but all may not vanish, for the function would then be finite 
everywhere in the parallelogram. 

It will however be seen, from the next succeeding propositions, that the 
division into classes is of most importance for functions of the second 
jrder. 

IV. Two functions, which are doubly-periodic in the same periods*, and 
which have the same zeros and the same infinities each in the same degrees 
respectively, are in a constant ratio. 

Let <f) and ^ be the functions, having the same periods; and let a of 
degree v, /3 of degree fi, ... be all the irreducible zeros of < and T/T; arid a of 
degree n, b of degree m, ... be all the irreducible infinities of <f> and of ty. 
Then a function G (z), without zeros or infinities within the parallelogram, 
exists such that 

, , , = (z-a) v (z-py ... G _ 

and another function H(z), without zeros or infinities within the parallelo 
gram, exists such that 



Hence *(*)_<?(*) 

- 



Now the function on the right-hand side has no zeros in the parallelogram, 
for G has no zeros and H has no infinities ; and it has no infinities in the 

* Such functions will be called homoperiodic. 



116.] OF THE SECOND ORDER 225 

parallelogram, for G has no infinities and H has no zeros : hence it has 
neither zeros nor infinities in the parallelogram. Since it is equal to the 
function on the left-hand side, which is a doubly-periodic function, it has no 
zeros and no infinities in the whole plane ; it is therefore a constant, say 
A. Thus* 



V. Two functions of the second order, doubly -periodic in the same periods 
and having the same infinities, are equivalent to one another. 

If one of the functions be of the first class in the second order, it has one 
irreducible double infinity, say at a ; so that we have 



where %(z) is finite everywhere within the parallelogram. Then the other 
function also has z = a for its sole irreducible infinity and that infinity is of 
the second degree ; therefore we have 

TT 



where ^ (z) is finite everywhere within the parallelogram. Hence 



Now x and % x are finite everywhere within the parallelogram, and therefore 
so is H% Gfo. But H% Gfo, being equal to the doubly-periodic function 
H(j) Gijr, is therefore doubly-periodic ; as it has no infinities within the 
parallelogram, it consequently can have none over the plane and therefore it 
is a constant, say 7. Thus 



proving that the functions <j> and ty are equivalent. 

If on the other hand one of the functions be of the second class in the 
second order, it has two irreducible simple infinities, say at 6 and c, so that 
we have 



where 6(z) is finite everywhere within the parallelogram. Then the other 
function also has z = b and z = c for its irreducible infinities, each of them 
being simple ; therefore we have 



where 6 l (z) is finite everywhere within the parallelogram. Hence 

(z) - Cty (z) = De (z) - Ce i (z}. 



* This proposition is the modified form of the proposition of 52, when the generalising 
exponential factor has been determined so as to admit of the periodicity. 

F. 15 



226 IRREDUCIBLE ZEROS [116. 

The right-hand side, being finite everywhere in the parallelogram, and equal 
to the left-hand side which is a doubly-periodic function, is finite everywhere 
in the plane ; it is therefore a constant, say B, so that 



proving that < and ty are equivalent to one another. 

It thus appears that in considering doubly-periodic functions of the second 
order, homoperiodic functions of the same class are equivalent to one another 
if they have the same infinities ; so that, practically, it is by their infinities 
that homoperiodic functions of the second order and the same class are dis 
criminated. 

COROLLARY 1. If two equivalent functions of tlie second order have one 
zero the same, all their zeros are the same. 

For in the one class the constant /, and in the other class the constant B, 
is seen to vanish on substituting for z the common zero ; and then the two 
functions always vanish together. 

COROLLARY 2. If two functions, doubly-periodic in the same periods but 
not necessarily of the second order, have the same infinities occurring in such a, j 
way that the fractional parts of the two functions are the same except as to a 
constant factor, the functions are equivalent to one another. And if, in 
addition, they have one zero common, then all their zeros are common, so 
that the functions are then in a constant ratio. 

COROLLARY 3. If two functions of the second order, doubly-periodic in( 
the same periods, have their zeros the same, and one infinity common, they are ^ 
in a constant ratio. 

VI. Every doubly -periodic function has as many irreducible zeros as it 
has irreducible infinities. 

Let < (z) be such a function. Then 



z +h z 

is a doubly-periodic function for any value of h, for the numerator is doubly- 
periodic and the denominator does not involve z ; so that, in the limit when 
h = 0, the function is doubly-periodic, that is, </> (z) is doubly-periodic. 

Now suppose <f>(z) has irreducible zeros of degree m 1 at a 1} ra 2 at a 2 , ..., 
and has irreducible infinities of degree /^ at 1} yu, 2 at 2 , ... ; so that the 
number of irreducible zeros is Wj + ra 2 + . . . , and the number of irreducible 
infinities is ^ 1 + /i 2 + ..., both of these numbers being finite. It has been 
shewn that < {z) can be expressed in the form 



116.] AND IRREDUCIBLE INFINITIES 227 

whore F(z) has neither a zero nor an infinity within, or on the boundary of, 
the parallelogram of reference. 

Since F(z) has a value, which is finite, continuous and different from zero 

Tjlt / \ 

everywhere within the parallelogram or on its boundary, the function -p4-r 

* W 
is not infinite within the same limits. Hence we have 



rr - ~ ... 

9 (z) za z 2 

+ -* + =*. + .. 

z ttj z a 2 

where g (z) has no infinities within, or on the boundary of, the parallelogram 
of reference. But, because <f> (z) and <f> (z) are doubly-periodic, their quotient 
is also doubly-periodic ; and therefore, applying Prop. II., we have 

m^ + w 2 + . . . ^ p 2 . . . = 0, 
that is, m 1 +m 2 + ... = fj,! + fi 2 + ..., 

or the number of irreducible zeros is equal to the number of irreducible 
infinities. 

COROLLARY I. The number of irreducible points for which a doubly - 
periodic function assumes a given value is equal to the number of irreducible 
zeros. 

For if the value be A, every infinity of $(z) is an infinity of the doubly- 
periodic function $ (z) A ; hence the number of the irreducible zeros of the 
latter is equal to the number of its irreducible infinities, which is the same as 
the number for < (z} and therefore the same as the number of irreducible 
zeros of < (z). And every irreducible zero of < (z} A is an irreducible 
point, for which < (z) assumes the value A. 

COROLLARY II. A doubly-periodic function with only a single zero does 
not exist; a doubly -periodic function of the second order has two zeros; and, 
generally, the order of a function can be measured by its number of irreducible 
zeros. 

Note. It may here be remarked that the doubly-periodic functions 
( 115), that have only accidental singularities in the finite part of the 
plane, have z = oo for an essential singularity. It is evident that for infinite 
values of z, the finite magnitude of the parallelogram of periods is not 
recognisable ; and thus for z = GO the function can have any value, shewing 
that z = oo is an essential singularity. 

VII. Let a 1} a 2) ... be the irreducible zeros of a function of degrees 
w 1; m 2 , ... respectively ; a 1} 2 , ... its irreducible infinities of degrees /^, /u, 2 , ... 
respectively; and z 1 ,z 2 ,... the irreducible points where it assumes a value c, 
which is neither zero nor infinity, their degrees being M 1} M. 2) ... respectively. 

152 



228 IRREDUCIBLE ZEROS [116. 

Then, except possibly as to additive multiples of Hie periods, the quantities 
2 m r a r , 2 UrCir and 2 M r z r are equal to one another, so that 

r=l r=l r=l 

2 m r a r = 2 M r z r = 2 p r ctr (mod. 2o>, 2&/)- 

r=l r=l r=l 

Let (/> (/) be the function. Then the quantities which occur are the sums 
of the zeros, the assigned values, and the infinities, the degree of each being 
taken account of when there is multiple occurrence ; and by the last 
proposition these degrees satisfy the relations 



The function <f)(z) c is doubly-periodic in 2u and 2&> ; its zeros are 
z 1} z. 2 , ... of degrees M 1} M^,... respectively; and its infinities are ct l , 2 , ... of 
degrees /i 1} yn 2 , , being the same as those of <(Y). Hence there exists a 
function G(z), without either a zero or an infinity lying in the parallelogram 
or on its boundary, such that </> 0) - c can be expressed in the form 

^l* 1 C.(*I* a >! " G (*) 

for all points not outside the parallelogram ; and therefore, for points in that 

region 

<f> (Y) ^ M r ^ *r G (z) 

\ /~** / \ 



/ \ 

<j)(z) C r =l Z Z r Z O. r 

Hence 

z$(z) ~ M r z v p r z zG (z) 

. . >. - 2< - --- 2* --- 1 .~ , . 
$(z) C r =l z z r Z a r W (*) 

= 2 Jf r+ 2 



, 

~r 



* ~r /-v / -. , 

=\Z Z r ZOL r (r(z) 

2 M r = 2 n r . 

r=l r=l 

Integrate both sides round the boundary of the fundamental parallelogram. 
Because G (z) has no zero and no infinity in the included region and does not 
vanish along the curve, the integral 

zG (z) 



I 



dz 



G(z) 

vanishes. But the points z { and 04 are enclosed in the area ; and therefore 
the value of the right-hand side is 

2iri 2 M r z r Ziri 2 /V*r, 



so 



that 



\Z) c 
the integral being extended round the parallelogram. 



116.] AND IRREDUCIBLE INFINITIES 229 

zd> (z) 

Denoting the subject of integration , by/(^), we have 

<p(z) c 

-/=*" - 



and therefore, by the Corollary to Prop. II., the value of the foregoing 
integral is 

* r -*-** r -* 

JA<f>(Z)-C J A (j)(z)-G 

the integrals being taken along the straight lines AD and AB respectively 
(fig. 33, p. 221). 

Let w <f)(z) c; then, as z describes a path, w will also describe a single 
path as it is a uniform function of z. When z moves from A to D, w moves 
from (j>(A)-c by some path to (f>(D) c, that is, it returns to its initial 
position since <f> (D) = <f> (A) ; hence, as z describes AD, w describes a simple 
closed path, the area included by which may or may not contain zeros and 
infinities of w. Now 

dw = <f> (z) dz, 

CD < ( z \ 

and therefore the integral I ,,\ dz is equal to 
* JAJ>(*)-C 



I 



dw 
w 



taken in some direction round the corresponding closed path for w. This 
integral vanishes, if no w-zero or w-infinity be included within the area 
bounded by the path ; it is + Im iri, if m be the excess of the number of 
included zeros over the number of included infinities, the + or sign being 
taken with a positive or a negative description ; hence we have 



where m is some positive or negative integer and may be zero. Similarly 



where n is some positive or negative integer and may be zero. 

Thus 27Ti (2,MrZ r ~ 2/V*,-) = 2w . 2w7n 2a) . Smri, 

and therefore ^M r z r ^p r i r = 2ma> 2?io> 

= (mod. 2&), 2o> ). 

Finally, since ^M r z r = 2/v*r whatever be the value of c, for the right-hand 



230 DOUBLY-PERIODIC FUNCTIONS [116. 

side is independent of c, we may assign to c any value we please. Let the 
value zero be assigned ; then ^M r z r becomes Sm r a r , so that 

^m r a r = "2/j, r f* r (mod. 2&>, 2&/). 

The combination of these results leads to the required theorem*, expressed 
by the congruences 

2 m r a r = 2 M r z r = 2 ^ r ^r (mod. 2o>, 2&> ). 

r=l r=l r=l 

Note. Any point within the parallelogram can be represented in the 
form z + a2&> + 62&> , where a and 6 are real positive quantities less than 

unity. Hence 

2 M r z r = A z 2a> + B z 2a> / + zM r , 

where J. and B are real positive quantities each less than 27lf r , that is, less 
than the order of the function. 

In particular, for functions of the second order, we have 

z 1 + z, = A z 2&> + B z 2&/ + 2.2-0, 
where A z and B z are positive quantities each less than 2. Similarly, if a and 

b be the zeros, 

a + b = A a 2w + a 2w + 2*o, 

where J. ffl and B a are each less than 2 ; hence, if 

^i + ^2 a b w2w + m 2o> , 

then w may have any one of the three values - 1, 0, 1 and so may m , the 
simultaneous values not being necessarily the same. 

Let a and ft be the infinities of a function of the second class ; then 
a + /3 a b = ?i2&) + n"2w , 

where n and ri may each have any one of the three values 1, 0, 1. By 
changing the origin of the fundamental parallelogram, so as to obtain a 
different set of irreducible points, we can secure that n and n are zero, 

and then 

a + @ = a+b. 

Thus, if n be 1 with an initial parallelogram, so that 

a + /3 = a + &+2&>, 

we should take either /3 - 2&> = {, or a - 2&> = a , according to the position of 
a and /3, and then have a new parallelogram such that 
a + @ = a + b, or a + ft = a + b. 

The case of exception is when the function is of the first class and has a 
repeated zero. 

* The foregoing proof is suggested by Konigsberger, Theorie der elliptischen Functionen, 
t. i, p. 342 ; other proofs are given by Briot and Bouquet and by Liouville, to whom the adopted 
form of the theorem is due. The theorem is substantially contained in one of Abel s general 
theorems in the comparison of transcendents. 



116.] OF THE SECOND ORDER 231 

VIII. Let $ (z) be a doubly -periodic function of the second order. If 7 
be the one double infinity when the function is of the first class, and if a and ft 
be the two simple infinities when the function is of the second class, then in the 
former case 



and in the latter case </> (z) < (a + (3 z). 

Since the function is of the second order, so that it has two irreducible 
infinities, there are two (and only two) irreducible points in a fundamental 
parallelogram at which the function can assume any the same value : let 
them be z and z . 

Then, for the first class of functions, we have 
z + z = 2 7 

= 27 + 2mo> + 2wa> , 

where m and n are integers ; and then, since <f)(z) = <j> (z } by definition of z 
and /, we have 

< (z) = < (27 - z + 2ma) 

= 0(27-4 
For the second class of functions, we have 

z + z = a. + /3 



so that, as before, 

(/> (z) = </> (a + /3 - z + 2ma) + 2wa> ) 



117. Among the functions which have the same periodicity as a given 
function </> (z), the one which is most closely related to it is its derivative 
< (z). We proceed to find the zeros and the infinities of the derivative of a 
function, in particular, of a function of the second order. 

Since (f> (z) is uniform, an irreducible infinity of degree n for </> (z) is an 
irreducible infinity of degree n -f 1 for (z). Moreover < (z), being uniform, 
has no infinity which is not an infinity of </> (z) ; thus the order of < (z) is 
2(?i + l) or its order is greater than that of cj>(z) by an integer which 
represents the number of distinct irreducible infinities of < (z), no account 
being taken of their degree. If, then, a function be of order m, the order of 
its derivative is not less than m + 1 and is not greater than 2m. 

Functions of the second order either possess one double infinity so that 
within the parallelogram they take the form 




and then <j> (z) = - + % (*), 



232 ZEROS OF THE DERIVATIVE [117. 

that is, the infinity of (f>(z) is the single infinity of tf> (z) and it is of the 
third degree, so that cf> (z) is of the third order ; or they possess two simple 
infinities, so that within the parallelogram they take the form 



and then f W = - G - - _ + x (,), 



that is, each of the simple infinities of < (z) is an infinity for </> (z) of the 
second degree, so that < (z) is of the fourth order. 

It is of importance (as will be seen presently) to know the zeros of 
the derivative of a function of the second order. 

For a function of the first class, let 7 be the irreducible infinity of the 
second degree ; then we have 



and therefore $ (2) = </> (^7 z). 

Now </> (z) is of the third order, having 7 for its irreducible infinity in the 
third degree : hence it has three irreducible zeros. 

In the foregoing equation, take z = 7 : then 

</> (7) = -$ (7), 

shewing that 7 is either a zero or an infinity. It is known to be the only 
infinity of < (z). 

Next, take z = 7 + &> ; then 

< (7 + &)) = $ (7 a>) 



= - < (7 + G>), 

shewing that 7 + &> is either a zero or an infinity. It is known not to be an 
infinity ; hence it is a zero. 

Similarly 7 + &/ and 7 + <u + &/ are zeros. Thus three zeros are obtained, 
distinct from one another ; and only three zeros are required ; if they be not 
within the parallelogram, we take the irreducible points homologous with 
them. Hence : 

IX. The three zeros of the derivative of a function, doubly -periodic in 
2eo and 2eo and having 7 for its double (and only) irreducible infinity, are 

7 + &), 7 + eo , 7 + w + w . 

For a function of the second class, let a and /3 be the two simple 
irreducible infinities; then we have 



and therefore <f> (z)= <f> (a + ft z). 



117.] OF A DOUBLY-PERIODIC FUNCTION 233 

Now (j) (z) is of the fourth order, having a and ft as its irreducible 
infinities each in the second degree ; hence it must have four irreducible 
zeros. 

In the foregoing equation, take z = \(VL + ft) ; then 



shewing that | (a + /3) is either a zero or an infinity. It is known not to be 
an infinity ; hence it is a zero. 

Next, take z = (a + (3) + w ; then 

f(}(t)+} --+ {*(+)-] 

= - < & ( a + ) - to + 2&>j 

.+ {*<+)+}, 

shewing that |(a + /3) + &> is either a zero or an infinity. As before, it is 
a zero. 

Similarly i (a + /3) + &> and i (a + /3) -f &> + &> are zeros. Four zeros are 
thus obtained, distinct from one another; and only four zeros are required. 
Hence : 

X. The four zeros of the derivative of a function, doubly-periodic in 2&> 
and 2o) and having a and /3 for its simple (and only) irreducible infinities, are 

i(a + /3), i(a + /3) + a>, i(a + /3) + ft> , |- (a + /3) + w + a/. 

The verification in each of these two cases of Prop. VII., that the sum of 
the zeros of the doubly-periodic function < (z) is congruent with the sum of 
its infinities, is immediate. 

Lastly, it may be noted that, if z l and z^ be the two irreducible points for 
which a doubly -periodic function of the second order assumes a given value, 
then the values of its derivative for z 1 and for z% are equal and opposite. For 

(j> (z) = <f> (a + /3 - z) = cf> (z, + z. 2 - z), 
since z l + z., = a + (3 ; and therefore 

<f> (z) = -$ (z, + z. 2 - z), 
that is, < (z l ) = </> (z 2 ), 

which proves the statement. 

118. We now come to a different class of theorems. 

XI. Any doubly -periodic function of the second order can be expressed 
algebraically in terms of an assigned doubly-periodic function of the second 
order, if the periods be the same. 

The theorem will be sufficiently illustrated and the line of proof 
sufficiently indicated, if we express a function (/> (z) of the second class, with 
irreducible infinities a, ft and irreducible zeros a, b such that a + (3 = a + b, in 



234 FUNCTIONS [118. 

terms of a function < of the first class with 7 as its irreducible double 
infinity. 

n .. , ,. 

Consider a function 



Q ( z + h ) _ 

A zero of <X> (z + h) is neither a zero nor an infinity of this function ; nor 
is an infinity of <1> (z + h) a zero or an infinity of the function. It will have 
a and 6 for its irreducible zeros, if 

a + h = h , 
b + h + h = 2 7 ; 

and these will be the only zeros, for <E> is of the second order. It will have o 
and yS for its irreducible infinities, if 



and these will be the only infinities, for < is of the second order. These 
equations are satisfied by 



Hence the assigned function, with these values of h, has the same zeros 
and the same infinities as $>(z); and it is doubly-periodic in the same periods. 
The ratio of the two functions is therefore a constant, by Prop. IV., so that 

c|> ( z + h) <I> (h ) 

If the expression be required in terms of <& (z) alone and constants, then 
<j> ( z 4. h} must be expressed in terms of <I> (z) and constants which are values 
of <X> (z) for special values of z. This will be effected later. 

The preceding proposition is a special case of a more general theorem 
which will be considered later ; the following is another special case of that 
theorem : viz. : 

XII. A doubly -periodic function with any number of simple infinities can 
be expressed either as a sum or as a product, of functions of the second order 
and the second class which are doubly-periodic in the same periods. 

Let j, 2 , ..., a n be the irreducible infinities of the function <, and 
suppose that the fractional part of <t> (z) is 

A-i , A 2 [ ^ i-+ ^n 

z ttj z 2 z a n 

with the condition A 1 + A 2 + + A n = Q. Let <j>n(z) be a function, 

doubly-periodic in the same periods, with a,-, a,- as its only irreducible infinities, 



118.] OF THE SECOND ORDER 235 

supposed simple; where i and j have the values 1, ,n. Then the 

fractional parts of the functions ^>j, (z), < 23 (z), . . . are 

0, 

G, 



i z a., 

I 
\z 2 ^ , 



respectively; and therefore the fractional part of 

^!^ / \ , 
W + 



is Al An - An ~ l 



z-a.! Z-CL, z-cin-T. z-a n 

Ai A n _^ A n 

= - -+...+- - + ^, 

Z-Cl! Z- _! Z - Ctn 



n 

since S -4* = 0. This is the same as the fractional part of <l> (z); and therefore 



- <^> 2 3 (f) - ... - -~ 



has no fractional part. It thus has no infinity within the parallelogram ; it 
is a doubly-periodic function and therefore has no infinity anywhere in the 
plane; and it is therefore merely a constant, say B. Hence, changing the 
constants, we have 

$>(z)-B^(z}-B.><t>v(z)-...-B n -,<t>n-,,n(z} = B, 

giving an expression for <$> (z} as a linear combination of functions of the 
second order and the second class. But as the assignment of the infinities is 
arbitrary, the expression is not unique. 

For the expression in the form of a product, we may denote the n 
irreducible zeros, supposed simple, by !,...,. We determine n - 2 new 
irreducible quantities c, such that 



C 2 = 



Cn2 &n\ ~r Cn3 ~ Q"ni > 
Cl n = tt n + C_ Q" 



n 

this being possible because 2 o^ = 2 a r ; and we denote by $ (z ; a, ft ; e, f) a 

=! r=l 

function of .gr, which is doubly-periodic in the periods of the given function, 



ALGEBRAICAL RELATIONS [118. 

has a and $ for simple irreducible infinities and has e and / for simple 
irreducible zeros. Then the function 

<f)(z; a l5 2 ; i, Ci) <f> ( z ; a s, c i ; 0-2, c 2 ) ...< (2 ; n , c n _ 2 ; a n _ l5 a n ) 
has neither a zero nor an infinity at c 1} at c 2 , ..., and at c n _ 2 ; it has simple 
infinities at a l} a 2 , ..., a n , and simple zeros at a lt a 2 , ..., a n - 1} a n . Hence it 
has the same irreducible infinities and the same irreducible zeros in the same 
degree as the given function < (z) ; and therefore, by Prop. IV., <I> (z) is 
a mere constant multiple of the foregoing product. 

The theorem is thus completely proved. 

Other developments for functions, the infinities of which are not simple, 
are possible ; but they are relatively unimportant in view of a theorem, 
Prop. XV., about to be proved, which expresses any periodic function in 
terms of a single function of the second order and its derivative. 

XIII. If two doubly -periodic functions have the same periods, they are 
connected by an algebraical equation. 

Let u be one of the functions, having n irreducible infinities, and v be 
the other, having m irreducible infinities. 

By Prop. VI., Corollary I., there are n irreducible values of z for a value 
of u; and to each irreducible value of z there is a doubly-infinite series of. 
values of z over the plane. The function v has the same value for all the 
points in any one series, so that a single value of v can be associated uniquely 
with each of the irreducible values of z, that is, there are n values of v for 
each value of u. Hence, ( 99), v is a root of an algebraical equation of the 
nth degree, the coefficients of which are functions of u. 

Similarly u is a root of an algebraical equation of the mth degree, the 
coefficients of which are functions of v. 

Hence, combining these results, we have an algebraical equation between 
u and v of the nth degree in v and the mth in u, where m and n are the 
respective orders of v and u. 

COROLLARY I. If both the functions be even functions of z, then n and m 
are even integers ; and the algebraical relation between u and v is of degree ^n 
in v and of degree ^m in u. 

COROLLARY II. If a function u be doubly-periodic in &> and &> , and a 
function v be doubly -periodic in fl and U , where 

n = mca + nta, I! = m w + nw! , 
m, n, m , n being integers, then there is an algebraic relation between u and v. 

119. It has been proved that, if a doubly-periodic function u be of order m, 
then its derivative du/dz is doubly-periodic in the same periods and is of an 
order n, which is not less than m + 1 and not greater than 2?/i. Hence, by 



119.] BETWEEN HOMOPERIODIC FUNCTIONS 237 

Prop. XIII., there subsists between u and u an algebraical equation of order m 
in u and of order n in u; let it be arranged in powers of u so that it takes 
the form 

U" u m _j_ JJ u mi _i _ _ i U _ u 2 i JJ _ u i JJ __ Q 

where U , U 1} ... , U m are rational integral algebraical functions of u one at 
least of which must be of degree n. 

Because the only distinct infinities of u are infinities of u, it is impossible 
that u should become infinite for finite values of u: hence U = can have no 
finite roots for u, that is, it is a constant and so it may be taken as unity. 

And because the m values of z, for which u assumes a given value, have 
their sum constant save as to integral multiples of the periods, we have 

corresponding to a variation 8u ; or 

du du du 

f/7/ 

Now is one of the values of u corresponding to the value of u, and so for 

the others ; hence 

3 1 



r =i u r 
that is, by the foregoing equation, 

" m i 



= 0, 



u n . 

and therefore U m -^ vanishes. Hence : 

XIV. There is a relation, between a doubly -periodic function u of order m 
and its derivative, of the form 

u m + U^ - 1 + ...+ U^u * + U m = 0, 

where U l} ..., U m _ 2 , U m are rational integral algebraical functions of u, at 
least one of which must be of degree n, the order of the derivative, and n is 
not less than m + 1 and not greater than 2m. 

Further, by taking v = - , which is a function of order m because it has the 

Uj 

m irreducible zeros of u for its infinities, and substituting, we have 

v f _ 03 U^ m ~ l + v*Uv m ~* - . . . v 2 " 1 - 4 U m _ 2 v 2 + v 2 " 1 U m = 0. 
The coefficients of this equation must be integral functions of v ; hence the 
degree of U r in u cannot be greater than 2r. 

COROLLARY. The foregoing equation becomes very simple in the case of 
doubly-periodic functions of the second order. 

Then m = 2. 



238 DIFFERENTIAL EQUATION [119. 

If the function have one infinity of the second degree, its derivative has 
that infinity in the third degree, and is of the third order, so that n = 3 ; and 
the equation is 

/y7 ?/ \2 

( ^ ) = \u? + 3/iw 2 + Svu + p, 
\d*J 

where X, /*, v, p are constants. If 6 be the infinity, so that 

A 
*.(,)_-_ + (*), 

where % (^) is everywhere finite in the parallelogram, then - = A ; and the 

/77/ 

zeros of -j- are 6 + o>, + &/, 6 + o> + CD ; so that 
diz 

a, )} { 



This is /ie general differential equation of Weierstrasss elliptic functions. 

If the function have two simple infinities a and @, its derivative has each 
of them as an infinity of the second degree, and is of the fourth order, so that 
n = 4 ; and the equation is 



(du\* _ 
(dz) = 



dM + c 2 w + >c 3 u + c 4 , 
where c , c 1} c 2 , c 3 , c 4 are constants. Moreover 



where ^ (^) is finite everywhere in the parallelogram. Then c u = G~ 2 ; and 

^/ i/ 

the zeros of -y- are ^ (a + /3), -|- (a + (3) + w, ^ (a -f /3) + co f , % (a + ft) + w + &> , 
ft/2 

so that the equation is 



( + 13)+ + }]. 
This is the general differential equation of Jacobis elliptic functions. 

The canonical forms of both of these equations will be obtained in Chapter 
XI., where some properties of the functions are investigated as special illustra 
tions of the general theorems. 

Note. All the derivatives of a doubly-periodic function are doubly- 
periodic in the same periods, and have the same infinities as the function but 
in different degrees. In the case of a function of the second order, which 
must satisfy one or other of the two foregoing equations, it is easy to see that 
a derivative of even rank is a rational, integral, algebraical function of u, and 
that a derivative of odd rank is the product of a rational, integral, algebraical 
function of u by the first derivative of u. 



119.] OF DOUBLY-PERIODIC FUNCTIONS 239 

It may be remarked that the form of these equations confirms the result 
at the end of 117, by giving two values of u for one value of u, the two 
values being equal and opposite. 

Ex. If u be a doubly-periodic function having a single irreducible infinity of the third 
degree so as to be expressible in the form 

2 6 
-o + -5 + integral function of z 

z z 

within the parallelogram of periods, then the differential equation of the first order which 
determines u is 



where 7 4 is a quartic function of u and where a is a constant which does not vanish with 6. 

(Math. Trip., Part II, 1889.) 

XV. Every doubly -periodic function can be expressed rationally in terms 
of a function of the second order, doubly-periodic in the same periods, and its 
derivative. 

Let u be a function of the second order and the second class, having the 
same two periods as v, a function of the rath order ; then, by Prop. XIII., 
there is an algebraical relation between u and v which, being of the second 
degree in v and the mth degree in u, may be taken in the form 

Lv* - 2Mv + P = 0, 

where the quantities L, M, P are rational, integral, algebraical functions of u 
and at least one of them is of degree m. Taking 

Lv-M=w, 

we have w 2 = M 2 LP, 

a rational, integral, algebraical function of u of degree not higher than 2w. 

Thus w cannot be infinite for any finite value of u : an infinite value of u 
makes w infinite, of finite multiplicity. To each value of u there correspond 
two values of w equal to one another but opposite in sign. 

Moreover w, being equal to Lv - M , is a uniform function of z, say F(z\ 
while it is a two-valued function of u. A value of u gives two distinct 
values of z, say z l and 2 ; hence the values of w, which arise from an assigned 
value of u, are values of w arising as uniform functions of the two distinct 
values of z. Hence as the two values of w are equal in magnitude and 
opposite in sign, we have 

r(4)+J*(4)-Oi 

that is, since ^ + z. 2 = a. + ft where a and /3 are the irreducible infinities of u, 



so that l(a + ), (a + /3) + a>, (a + ) + , and (a + /3) + a> + a> are either 
zeros or infinities of w. They are known not to be infinities of u, and w is 
infinite only for infinite values of u ; hence the four points are zeros of w. 



240 RELATIONS BETWEEN [119. 

But these are all the irreducible zeros of u ; hence the zeros of u are 
included among the zeros of w. 

Now consider the function w/u . The numerator has two values equal 
and opposite for an assigned value of u ; so also has the denominator. Hence 
w/u is a uniform function of u. 

This uniform function of u may become infinite for 
(i) infinities of the numerator, 
(ii) zeros of the denominator. 

But, so far as concerns (ii), we know that the four irreducible zeros of the 
denominator are all simple zeros of u and each of them is a zero of w .; hence 
w/u does not become infinite for any of the points in (ii). And, so far as 
concerns (i), we know that all of them are infinities of u. Hence w/u, a 
uniform function of u, can become infinite only for an infinite value of u, and 
its multiplicity for such a value is finite; hence it is a rational, integral, 
algebraical function of u, say N, so that 

w = Nu . 

Moreover, because w 2 is of degree in u not higher than 2m, and u 2 is of 
the fourth degree in u, it follows that N is of degree not higher than m 2. 

We thus have Lv M Nu, 

M+Nu M N , 

v= ~r = L + L U > 

where L, M, N are rational, integral, algebraical functions of u ; the degrees 
of L and M are not higher than m, and that of N is not higher than m 2. 

Note 1. The function u, which has been considered in the preceding 
proof, is of the second order and the second class. If a function u of the 
second order and the first class, having a double irreducible infinity, be 
chosen, the course of proof is similar ; the function w has the three irreducible 
zeros of u among its zeros and the result, as before, is 

w = Nu . 

But, now, w"- is of degree in u not higher than 2m and u 2 is of the third 
degree in u ; hence N is of degree not higher than m 2 and the degree of w 2 
in u cannot be higher than 2m 1. 

Hence, if L, M, P be all of degree m, the terms of degree 2m in LP M 2 
disappear. If all of them be not of degree m, the degree of M must not be 
higher than m l ; the degree of either L or P must be m, but the degree 
of the other must not be greater than ml, for otherwise the algebraical 
equation between u and v would not be of degree m in u. 

We thus have 

Lv 2 - 2Mv + P = (), Lv - M = Nu , 



119.] HOMOPERIODIC FUNCTIONS 241 

where the degree of N in u is not higher than m 2. If the degree of L be 
less than TO, the degree of M is not higher than TO 1 and the degree of P is 
TO. If the degree of L be m, the degree of M may also be m provided that the 
degree of P be TO and that the highest terms be such that the coefficient 
of u 2m in LP - M 2 vanishes. 

Note 2. The theorem expresses a function v rationally in terms of u and 
u : but u is an irrational function of u, so that v is not expressed rationally 
in terms of u alone. 

But, in Propositions XI. and XII., it was indicated that a function such as 
v could be rationally expressed in terms of a doubly-periodic function, such as 
u. The apparent contradiction is explained by the fact that, in the earlier 
propositions, the arguments of the function u in the rational expression and 
of the function v are not the same ; whereas, in the later proposition whereby 
v is expressed in general irrationally in terms of u, the arguments are the 
same. The transition from the first (which is the less useful form) to the 
second is made by expressing the functions of those different arguments in 
terms of functions of the same argument when (as will appear subsequently, in 
121, in proving the so-called addition-theorem) the irrational function of u, 
represented by the derivative u, is introduced. 

COROLLARY I. Let H denote the sum of the irreducible infinities or of 
the irreducible zeros of the function u of the second order, so that H = 2y for 
functions of the first class, and O = a + /3 for functions of the second class. 
Let u be represented by <f> (z) and v by ty (z), when the argument must be put 
in evidence. Then 



so that W- Z ) = 

J-j i_j j 

Hence ^ (z) + ^ (fl - z) = 2 ^= 2R, 

JL 



First, if y (z) = ,Jr (ft - z\ then S = and ^ (z) = R : that is, a function ^ (z), 
which satisfies the equation 



can be expressed as a rational algebraical meromorphic function of <f> (z) of the 
second order, doubly -periodic in the same periods and having the sum of its 
irreducible infinities congruent with O. 

Second, if ^ ( e ) = - y, (fl _ z \ then R = and ^ (*) = flf (*) ; that is, 
function ^ (z), which satisfies the equation 



16 



a 



242 HOMOPERIODIC FUNCTIONS [119. 

can be expressed as a rational algebraical meromorphic function of < (z), 
multiplied by (z), where $ (z} is doubly-periodic in the same periods, is of the 
second order, and has the sum of its irreducible infinities congruent with Q. 

Third, if ty(z) have no infinities except those of u, it cannot become 
infinite for finite values of u ; hence L = has no roots, that is, L is a constant 
which may be taken to be unity. Then i/r (z) a function of order m can be 
expressed in the form 



where, if the function </> (z) be of the second class, the degree of M is not 
higher than m ; but, if it be of the first class, the degree of M is not higher 
than m - 1 ; and in each case the degree of N is not higher than m - 2. 

It will be found in practice, with functions of the first class, that these 
upper limits for degrees can be considerably reduced by counting the degrees 
of the infinities in 



Thus, if the degree of M in u be ^ and of N be \ the highest degree of an : 
infinity is either 2/t or 2X + 3 ; so that, if the order of ^ (z) be m, we should 

have 

m = 2/j, or m = 2\ + 3, > 

according as m is even or odd. 

When functions of the second class are used to represent a function ^r (z), 
which has two infinities a and /3 each of degree n, then it is easy to see that 
M is of degree n and N of degree n - 2 ; and so for other cases. 

COROLLARY II. Any doubly -periodic function can be expressed rationally 
in terms of any other function u of any order n, doubly-periodic in the same 
periods, and of its derivative ; and this rational expression can always be taken 
in the form 

U + U,U + t/X 3 + + Un-,u n ~\ 

where U , ... , 7 n -i are algebraical, rational, meromorphic functions of u. 

COROLLARY III. If <f) be a doubly-periodic function, then <f> (u + v) can be 
expressed in the form 



where ^ is a doubly -periodic function in the same periods and of the second 
order : each of the functions A, D, E is a symmetric function of^(u) and i/r (v), 
and B is the same function of^(v) and ty(u) as C is of ty (u) and ty (v). 

The degrees of A and E are not greater than m in ty ( u ) and than m in ^ (v), 
where m is the order of </> ; the degree of D is not greater than m - 2 in ^ (u) 
and than m - 2 in ^ (v) ; the degree of B is not greater than m - 2 in ^ (u) 
and than m in ^ (v), and the degree of C is not greater than m - 2 in -^ (v) 
and than m in -^ (u). 



CHAPTER XI. 

DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER. 

THE present chapter will be devoted, in illustration of the preceding 
theorems, to the establishment of some of the fundamental formulae relating 
to doubly-periodic functions of the second order which, as has already (in 
119, Cor. to Prop. XIV.) been indicated, are substantially elliptic functions : 
but for any development of their properties, recourse must be had to treatises 
on elliptic functions. 

It may be remarked that, in dealing with doubly-periodic functions, we 
may restrict ourselves to a discussion of even functions and of odd functions. 
For, if (/> (z) be any function, then {<j> (z} + <j>( z}} is an even function, and 
\ {(f>(z) </>( z}} is an odd function, both of them being doubly-periodic in 
the periods of <f> (z) ; and the new functions would, in general, be of order 
double that of <J>(z). We shall practically limit the discussion to even 
functions and odd functions of the second order. 

120. Consider a function <j>(z\ doubly-periodic in 2&> and 2w ; and let 
it be an odd function of the second class, with a and ft as its irreducible 
infinities, and a and b as its irreducible zeros*. 

Then we have < (z) = (f> (a + /3 z) 

which always holds, and <f> ( z) = </> (z) 

which holds because < (z) is an odd function. Hence 

<f> (a + /3 + z) = (/>(- *) 

= -$(*) 

so that a + ft is not a period ; and 



-*(*), 

To fix the ideas, it will be convenient to compare it with snz, for which 2w = 4^T, 2<a = 
a=iK , p=iK + 2K, a-0, and b = 2K. 

162 



244 DOUBLY-PERIODIC FUNCTIONS [120. 

whence 2 (a + /S) is a period. Since a -f /3 is not a period, we take a + /3 = a>, 
or = &) , or = &> + w ; the first two alternatives merely interchange &> and &> , so 

that we have either 

a + /3 = o), 

or a + /3 = ft) + &> . 

And we know that, in general, 

a + b = a + /3. 
First, for the zeros : we have 



so that </>(0) is either zero or infinite. The choice is at our disposal; for 

- satisfies all the equations which have been satisfied by $(z) and an 

</>(*) 

infinity of either is a zero of the other. We therefore take 



so that we have a = 0, 

6 = to or &) + ft) . 
Next, for the infinities : we have 

*(*)$(-*) 

and therefore <j> (- a) = - $ (a) = oo . 

The only infinities of < are a and /3, so that either 

a= a, 

or -CL = P. 

The latter cannot hold, because it would give a + /3 = whereas 

or = &> + &/; hence 

2a = 0, 

which must be associated with a + /3 = w or with a + /3 = &> + &/. 

Hence a, being a point inside the fundamental parallelogram, is either 0, 
a), &) , or tw + &) . 

It cannot be in any case, for that is a zero. 

If a _|_ ^ = Wj then a cannot be tw, because that value would give ft = 0, 
which is a zero, not an infinity. Hence either a = , and then /3 = &/ + &>; 
or a = &) + &), and then /3 = ft) . These are effectively one solution ; so that, if 

a + /3 = &), we have 

a, /3 = ft) , &> + &)) 

and a, 6 = 0, &) ) 

jf a + /S = w + &> , then a cannot be CD f &) , because that value would give 
{$ = 0, which is a zero, not an infinity. Hence either a = &> and then ft = &) , 
or a = ft) and then /3 = &). These again are effectively one solution ; so that, 

if a + /3 = &) + &> , we have 

a, = o), ft) 
and a, 6 = 0, o + ft) ) 



120.] OF THE SECOND CLASS 245 

This combination can, by a change of fundamental parallelogram, be made 
the same as the former ; for, taking as new periods 

2ft/ = 2a> t 2fl = 2 + 2a> , 
which give a new fundamental parallelogram, we have a + j3 = H, and 

a, ft &> , ft ft/, that is, ft/, ft 03 + 2<o 
so that a, /3 = ft/, O + a/] 

and a, b = 0, 

being the same as the former with O instead of &>. Hence it is sufficient to 
retain the first solution alone : and therefore 

a = to , ft = CD + co, 
a = 0, 6 = w. 

Hence, by 116, 1., we have 



where F(z) is finite everywhere within the parallelogram. 

Again, $ (z + a/) has z = and z = &> as its irreducible infinities, and 
it has 2 = 0) and z = &> + &/ as its irreducible zeros, within the parallelogram 
of (f) (z} ; hence 



where ^ (2) is finite everywhere within the parallelogram. Thus 



a function which is finite everywhere within the parallelogram ; since it is 
doubly-periodic, it is finite everywhere in the plane and it is therefore a 
constant and equal to the value at any point. Taking - i&/ as the point 
(which is neither a zero nor an infinity) and remembering that </> is an odd 
function, we have 

* (*)*(* + = - ft (* )} = p 

k being a constant used to represent the value of - {< (^o/)}" 2 . 

Also <j>(z + o)) = <f>(z + a + /3- 2&/) 

= c/>0 + a + /3)=-( (z), 
and therefore also < (&> z) = <f) (z). 

The irreducible zeros of <j> (z) were obtained in 117, X. In the 
present example, those points are a> + ft>, &> + ffc>, &>, f &> ; so that, as 
there, we have 

to ( W-{*(i)-4>(i*yito(*)-HW 

where K is a constant. But 

$ (f) = (2 - lft>) = (f) (-!) = _( (1 a,) ; 



246 DOUBLY-PERIODIC FUNCTIONS [120. 

and 0(fw + &/) = <(2a> + 2w -.!&>- a/) 

= <(- 2 <o -to ) 

= </>(&> + &> ); 

so that ,. . 4 I - 



where J. is a new constant, evidently equal to {< (0)} 2 . Now, as we know 
the periods, the irreducible zeros and the irreducible infinities of the function 
</> (z), it is completely determinate save as to a constant factor. To determine 
this factor we need only know the value of <$>(z) for any particular finite 
value of z. Let the factor be determined by the condition 



then, since <(^ft>)<(^G> + ft/) = T 

by a preceding equation, we have 



and then 

ft (*)} - {f (0)} [1 - {< (*)} 2 ] [1 - fr {(/> (*)} ] 



Hence, since (/> (2) is an odd function, we have 

< (z) = sn (//,). 

Evidently 2/xtu, 2/^ft) = 4^T, 2^ , where K and ^T have the ordinary signifi 
cations. The simplest case arises when /A = 1. 

121. Before proceeding to the deduction of the properties of even 
functions of z which are doubly-periodic, it is desirable to obtain the 
addition-theorem for <f>, that is, the expression of <p (y + z) in terms of 
functions of y alone and z alone. 

When <f> (y + z) is regarded as a function of z, which is necessarily of the 
second order, it is ( 119, XV.) of the form 



where M and L are of degree in < (z) not higher than 2 and N is independent 
of z. Moreover y + z = a and y + z = ft are the irreducible simple infinities 
of <j) (y + z) ; so that L, as a function of z, may be expressed in the form 



and therefore 

Z_^(iHL^^^)}l 
(z) - 



121.] OF THE SECOND CLASS 247 

where P, Q, R, S are independent of z but they may be functions of y. Now 

</> (a - y) = </> (w - y) = - 



and < (/3 y) = <j> (&> + w y} = 



so that the denominator of the expression for <f> (y + 2) is 



Since </> (z) is an odd function, < (#) is even ; hence 

, A p - 
~ */ 



and therefore $ (y + z) $ (y z) = - - 



Differentiating with regard to z and then making z = 0, we have 



so that, substituting for Q we have 



Interchanging y and z and noting that </> (t/ z) = (f) (z y), we have 



md therefore d> C7y * Z } d> (0} - 

W+*)1>< 



which is the addition-theorem required. 

Ex. If f(u) be a doubly-periodic function of the second order with infinities 6 1} i 2 , 
and 0(tt) a doubly -periodic function of the second order with infinities a lt a 2 such that, 
in the vicinity of (for i 1, 2), we have 

^ ( M ) = ,7~!r +Pi+& ( u ~ a i) + ...... > 

c6 u-j 

thon /M-/W = i W+* W-ft-ftl- 

the periods being the same for both functions. Verify the theorem when the functions are 
sn u and sn (u + v}. (Math. Trip. Part II., 1 891.) 

Prove also that, for the function $ (u), the coefficients p and p 2 are equal. (Burnside.) 

122. The preceding discussion of uneven doubly-periodic functions 
having two simple irreducible infinities is a sufficient illustration of the 



248 DOUBLY-PERIODIC FUNCTIONS [122. 

method of procedure. That, which now follows, relates to doubly- periodic 
functions with one irreducible infinity of the second degree ; and it will be 
used to deduce some of the leading properties of Weierstrass s er-function 
(of 57) and of functions which arise from it. 

The definition of the <r-function is 



where fi = 2ma> + 2m a) , the ratio of &> : &> not being purely real, and the 
infinite product is extended over all terms that are given by assigning to 
m and to m all positive and negative integral values from +00 to oo , 
excepting only simultaneous zero values. It has been proved (and it is 
easy to verify quite independently) that, when cr(z) is regarded as the 
product of the primary factors 



the doubly-infinite product converges uniformly and unconditionally for all 
values of z in the finite part of the plane ; therefore the function which it 
represents can, in the vicinity of any point c in the plane, be expanded in a 
converging series of positive powers of z c, but the series will only express 
the function in the domain of c. The series, however, can be continued over 
the whole plane. 

It is at once evident that a- (z) is not a doubly-periodic function, for it has 
no infinity in any finite part of the plane. 

It is also evident that a (z) is an odd function. For a change of sign in z 
in a primary factor only interchanges that factor with the one which has 
equal and opposite values of m and of m , so that the product of the two factors 
is unaltered. Hence the product of all the primary factors, being independent 
of the nature of the infinite limits, is an even function ; when z is associated 
as a factor, the function becomes uneven and it is a- (z). 

The first derivative, a (z), is therefore an even function ; and it is not 
infinite for any point in the finite part of the plane. 

It will appear that, though a- (z) is not periodic, it is connected with 
functions that have 2o> and 2&> for periods ; and therefore the plane will be 
divided up into parallelograms. When the whole plane is divided up, as in 
105, into parallelograms, the adjacent sides of which are vectorial repre 
sentations of 2w and 2&/, the function a-(z) has one, and only one, zero in 
each parallelogram; each such zero is simple, and their aggregate is given 
by z = l. The parallelogram of reference can be chosen so that a zero 
of <r (z} does not lie upon its boundary ; and, except where explicit account is 



122.] OF THE FIRST CLASS 249 

taken of the alternative, we shall assume that the argument of &> is greater 
than the argument of to, so that the real part* of w /ia) is positive. 

123. We now proceed to obtain other expressions for a- (z), and particu 
larly, in the knowledge that it can be represented by a converging series in 
the vicinity of any point, to obtain a useful expression in the form of a series, 
converging in the vicinity of the origin. 

Since er (z) is represented by an infinite product that converges uniformly 
and unconditionally for all finite values of z, its logarithm is equal to the sum 
of the logarithms of its factors, so that 



where the series on the right-hand side extends to the same combinations of 
m and m as the infinite product for z, and, when it is regarded as a sum of 

z z^ ( z\ 

functions o + i 7^2 + ^ ( ^ ~ r> ) ^ ne sei> i es converges uniformly and uncon- 

__ -- \ 1 - , 

ditionally, except for points z = l. This expression is valid for log a (z) over 
the whole plane. 

Now let these additive functions be expanded, as in 82. In the imme 
diate vicinity of the origin, we have 



a series which converges uniformly and unconditionally in that vicinity. 
Then the double series in the expression for log a (z} becomes 



and as this new series converges uniformly and unconditionally for points in 
the vicinity of z = 0, we can, as in 82, take it in the form 

oo ~r ( oo oo } 

5" J 5* y O-n 

4 \ ^-> <5r () 

r=3 r (-00 -oo J 

which will also, for such values of z, converge uniformly and unconditionally. 
In 56, it was proved that each of the coefficients 

00 00 

2 s n-*-, 



00 - 00 



for r = 3, 4,..., is finite, and has a value independent of the nature of the 
infinite limits in the summation. When we make the positive infinite limit 
for m numerically equal to the negative infinite limit for m, and likewise for 



This quantity is often denoted by ffi ( . - J . 



250 WEIERSTRASS S [123. 

ra , then each of these coefficients determined by an odd index r vanishes, 
and therefore it vanishes in general. We then have 

log a- 0) = log z - I* 22ft- 4 - ^ 22ft- 6 - ^ 22ft- 8 

a series which converges uniformly and unconditionally in the vicinity of the 
origin. 

The coefficients, which occur, involve o and , two independent constants. 
It is convenient to introduce two other magnitudes, g. 2 and g 3 , denned by the 

equations 

# 2 = 6022ft- 4 , #3 = 140220-0, 

so that g 2 and </ 3 are evidently independent of one another; then all the 
remaining coefficients are functions* of g. 2 and g 3 . We thus have 



and therefore <r (z) = ze m 

where the series in the index, containing only even powers of z, converges 

uniformly and unconditionally in the vicinity of the origin. 

It is sufficiently evident that this expression for a- (z) is an effective 
representation only in the vicinity of the origin ; for points in the vicinity of 
any other zero of cr (z), say c, a similar expression in powers of z - c instead 
of in powers of z would be obtained. 

124. From the first form of the expression for log cr (z), we have 



o-(z) z _, _ 

where the quantity in the bracket on the right-hand side is to be regarded as 
an element of summation, being derived from the primary factor in the 
product-expression for cr (z\ 



We write (z) = , ^ , 

so that %(z) is, by 122, an odd function, a result also easily derived from the 
foregoing equation ; and so 



This expression for (z) is valid over the whole plane. 
Evidently (z) has simple infinities given by 

for all values of ra and of m between + oo and - oo , including simultaneous 
zeros. There is only one infinity in each parallelogram, and it is simple ; for 
the function is the logarithmic derivative of a (z\ which has no infinity and 

* See Quart. Journ., vol. xxii., pp. 4, 5. The magnitudes g 2 and g 3 are often called the 
invariants. 



124.] ELLIPTIC FUNCTION 251 

only one zero (a simple zero) in the parallelogram. Hence %(z) is not a 
doubly-periodic function. 

For points, which are in the immediate vicinity of the origin, we have 



but, as in the case of cr(z), this is an effective representation of %(z) only 
in the vicinity of the origin ; and a different expression would be used for 
points in the vicinity of any other infinity. 

We again introduce a new function g> (z) defined by the equation 



Because is an odd function, $ (z) is an even function ; and 



where the quantity in the bracket is to be regarded as an element of 
summation. This expression for $ (z) is valid over the whole plane. 
Evidently |p (z) has infinities, each of the second degree, given by z = fl, 
for all values of m and of m between -f oo and - oo , including simultaneous 
zeros ; and there is one, and only one, of these infinities in each parallelogram. 
One of these infinities is the origin; using the expression which represents 
log a- (z) in the immediate vicinity of the origin, we have 



= - 2 + 20 9** + ^ 9*?+ 

for points z in the immediate vicinity of the origin. A corresponding 
expression exists for g> (z) in the vicinity of any other infinity. 

125. The importance of the function $ (z) is due to the following 
theorem : 

The function $> (z) is doubly-periodic, the periods being 2<w and 2&/. 
Wo have -l 



where the doubly-infinite summation excludes simultaneous zero values, and 
the expression is valid over the whole plane. Hence 

+ ^-n - Si 



252 WEIERSTRASS S [125. 

so that 



obtained by combining together the elements of the summation in g> (z + 2<w) 
and |p (,z). The two terms, not included in the summation, can be included, 
if we remove the numerical restriction as to non-admittance of simultaneous 
zero values for m and m \ and then 

2.) - f (,) = 2 



_ 



where now the summation is for all values of m and of m from + oo to oo . 
Let q denote the infinite limit of m, and p that of m . Then terms in the 
first fraction, for = 2 (mm + m w }, are the same as terms in the second for 
1 = 2 (m l)w + 2m w ; cancelling these, we have 

m =p 

-fC = 



where q is infinite. But 



?r) 2 sin 2 c 
and therefore 

= p = oo 1 ^.2 

2i 



- 2mV} 2 W . 
sin 






2o/ 

if/) be infinitely great compared with q. This condition may be assumed for 
the present purpose, because the value of g> (z) is independent of the nature 
of the infinite limits in the summation and is therefore unaffected by such a 
limitation. 

f - "" 1 ] f +! ? (9+1) -*$-* * 

l_* J l_ 

The fraction , has a real part. In the exponent it is multiplied by q + 1. 

that is, by an infinite quantity ; so that the real part of the index of 
the exponential is infinite, either positive or negative. Thus either the 
first term is infinite and the second zero, or vice versa; in either case, 

r T i 

sin \z + 2 (q + 1) twl ^ , is infinite, and therefore 
2o) J 



{ 2 + 2(q + l)(o- 2m V} 2 
Similarly for the other sum. Hence 



= 0. 



In the same way it may be shewn that 

>0 + 2a/)->0) = 0; 

therefore > (z) is doubly-periodic in 2<o and 2a> . 



126.] ELLIPTIC FUNCTION 253 

Now in any parallelogram whose adjacent sides are 2&> and 2&> , there is 
only one infinity and it is of multiplicity two; hence, by 116, Prop. III., 
Cor. 3, 2o) and 2&> determine a primitive parallelogram for $> (z). 

We shall assume the parallelogram of reference chosen so as to include 
the origin. 

126. The function $ (z) is thus of the second order and the first class. 

Since its irreducible infinity is of the second degree, the only irreducible 
infinity of g> (z} is of the third degree, being the origin ; and the function 
<t (z) is odd. 

The zeros of jp (z} are thus &>, ft/, and (&> + to ) ; or, if we introduce a new 
quantity w" defined by the equation 

&>" = &) + &> , 

the zeros of <@! (z) are &>, &> , &>". 
We take 

#>() = e 1} p(a>") = e 2 , p( m ) = e 3 , %>(z) = p: 
and then, by 119, Prop. XIV., Cor., we have 



where A is some constant. To determine the equation more exactly, we 
substitute the expression of jp in the vicinity of the origin. Then 



80 that P = -j+iQff* + 

When substitution is made, it is necessary to retain in the expansion all 
terms up to z inclusive. We then have, for |p 2 , the expression 

4 2 4 



and for A (^ - e-,) (p - e 2 ) (p - e 3 ), the expression 

A r 1 3 9- 3 

1 L^ 6 + 20^ + 28 5r3+> " 

- (e, + e 2 + e 3 )(^+g 2 + ...)+ ( 6l e, + e 2 e 3 + e&) (- + ...)- tf,A | 

When we equate coefficients in these two expressions, we find 



e 1 + e 2 + e 3 = 0, e& + e 
therefore the differential equation satisfied by p is 



254 PERIODICITY [126. 

Evidently >" = 6> 2 - %g s , 



and so on ; and it is easy to verify that the 2wth derivative of g) is a rational 
integral algebraical function of <p of degree n + l and that the (2w+l)th 
derivative of fp is the product of g> by a rational integral algebraical function 
of degree n. 

The differential equation can be otherwise obtained, by dependence on 
Cor. 2, Prop. V. of 116. We have, by differentiation of %> , 



for points in the vicinity of the origin ; and also 

^ + !^ 2 + r4^ 2 + "-- 

Hence <@" and > 2 have the same irreducible infinities in the same degree and 
their fractional parts are essentially the same : they are homoperiodic and 
therefore they are equivalent to one another. It is easy to see that g>" 6(jp 2 
is equal to a function which, being finite in the vicinity of the origin, is finite 
in the parallelogram of reference and therefore, as it is doubly -periodic, is 
finite over the whole plane. It therefore has a constant value, which can be 
obtained by taking the value at any point; the value of the function for 

z = is \g and therefore 

g>"_6^ = -^ 2 , 

so that |p"= 6g 2 -|<7 2 , 

the integration of which, with determination of the constant of integration, 

leads to the former equation. 

This form, involving the second derivative, is a convenient one by which 
to determine a few more terms of the expansion in the vicinity of the origin : 
and it is easy to shew that 



from which some theorems relating to the sums ^SH" 2 * 1 can be deduced*. 

Ex. If c n be the coefficient of 2 2n ~ 2 in the expansion of $ (z) in the vicinity of the 
origin, then 

c =/o~ . iw.. ON 2 Crfin-r- (Weierstrass.) 



We have jp 2 = 4^> 3 - g$ - g 3 ; 

the function jjp is odd and in the vicinity of the origin we have 



* See a paper by the author, Quart. Journ., vol. xxii, (1887), pp. 1 43, where other references 
are given and other applications of the general theorems are made. 



126.] OF WEIERSTRASS S FUNCTION 255 

hence, representing by (4|p 3 g$> g^ that branch of the function which is 
negative for large real values, we have 



and therefore z = 

The upper limit is determined by the fact that when z = 0, g> = oo ; so that 

- r d 

_ r d%> 

lp {4 (p - ej (p - e 2 ) (p - e,)}* 

This is, as it should be, an integral with a doubly-infinite series of values. 
We have, by Ex. 6 of 104, 

r 

0)j = ft) =| 

J<h 



, 

ft> 3 = ft) = 

J 

with the relation a)" = a) + co . 

127. We have seen that g> (z) is doubly-periodic, so that 

p(*+2) = $>(*), 

and therefore dg(5 + 2) = dgW 

a^ dz 

hence integrating ?(^ + 2<) = %(z) + A. 

Now ^ is an odd function ; hence, taking z co which is not an infinity of , 
we have 

^ = 2^(&))=27; 

say, where r) denotes (&>) ; and therefore 

(*-* 2)r- (*) 89, 

which is a constant. 

Similarly %(z + 2&> ) - ^ (^) - 2i/, 

where r; = ^ (to ) and is constant. 

Hence combining the results, we have 

% (z + 2w&) + 2?rc V) -(z) = 2mri + Zmrj , 
where m and m are any integers. 

It is evident that 77 and rj cannot be absorbed into ; so that is not a 
periodic function, a result confirmatory of the statement in 124. 



256 PSEUDO-PERIODIC [127. 

There is, however, a pseudo-periodicity of the function : its characteristic 
is the reproduction of the function with an added constant for an added 
period. This form is only one of several simple forms of pseudo-periodicity 
which will be considered in the next chapter. 

128. But, though %(z) is not periodic, functions which are periodic can 
be constructed by its means. 

Thus, if 4>(z)=AS(z-a)+Bt(z-V) 
then * + 2w-(*) = 2A(*-a 



so that, subject to the condition 

A+B+C+...=0, 
<j) (z) is a doubly-periodic function. 

Again, we know that, within the fundamental parallelograTH, f has a 
single irreducible infinity and that the infinity is simple; hence the irre 
ducible infinities of the function </> (V) are z = a, b, c, ..., and each is a simple 
infinity. The condition A + B + C + ...=0 is merely the condition of Prop. 
III., 116, that the integral residue of the function is zero. 

Conversely, a doubly-periodic function with m assigned infinities can be 
expressed in terms of f and its derivatives. Let a x be an irreducible infinity 
of <> of degree n, and suppose that the fractional part of <I> for expansion in 
the immediate vicinity of a x is 

A! i?i | ^ KI 

Then 

-if (*- 4).-... 



is not infinite for z = a^. 

Proceeding similarly for each of the irreducible infinities, we have a 
function 



r 

which is not infinite for any of the points z = a lt a 2 , .... But because <f> (z) 
is doubly-periodic, we have 

and therefore the function 



128.] 



FUNCTIONS 



257 



is doubly-periodic. Moreover, all the derivatives of any order of each of the 
functions are doubly-periodic; hence the foregoing function is doubly- 
periodic. 

The function has been shewn to be not infinite at the points a 1} a 2 , ..., 
and therefore it has no infinities in the fundamental parallelogram ; con 
sequently, being doubly-periodic, it has no infinities in the plane and it is 
a constant, say G. Hence we have 



g, 



r=i 



r =i 



m 

with the condition 2 A r = 0, which is satisfied because <E> (z) is doubly- 
periodic. 

This is the required expression * for <I> (z) in terms of the function % and 
its derivatives; it is evidently of especial importance when the indefinite 
integral of a doubly-periodic function is required. 

129. Constants 77 and 77 , connected with &> and , have been introduced 
by the pseudo-periodicity of (z)\ the relation, contained in the following 
proposition, is necessary and useful : 

The constants 77, w , &>, &> are connected by the relation 



the + or - sign being taken according as the real part of o> fa)i is positive or 
negative. 

A fundamental parallelogram having an angular point at z is either of 

the form (i) in fig. 34, in which case 9t f-^] is 

\mj 

positive : or of the form (ii), in which case 9J ( . ) 

\Ct)l/ 

is negative. Evidently a description of the paral 
lelogram A BCD in (i) will give for an integral the 
same result (but with an opposite sign) as a de 
scription of the parallelogram in (ii) for the same 
integral in the direction A BCD in that figure. 

We choose the fundamental parallelogram, so 
that it may contain the origin in the included 
area. The origin is the only infinity of which 
can be within the area : along the boundary is 
always finite. 

Now since 

* See Hermitet Ann. de Toulouse, t. ii, (1888), C, pp. 112. 
F. 




2 +2o) 



Fig. 34 



258 PSEUDO-PERIODICITY OF WEIERSTRASS s [129. 

the integral of (z) round ABCD in (i), fig. 34, is ( 116, Prop. II., Cor.) 

rD CB 

2r)dz - fy dz, 

J A J A 

the integrals being along the lines AD and AB respectively, that is, the 

integral is 

4 (rju> rfw}. 

But as the origin is the only infinity within the parallelogram, the path of 
integration ABCD A can be deformed so as to be merely a small curve round ! 
the origin. In the vicinity of the origin, we have 



and therefore, as the integrals of all terms except the first vanish when taken 
round this curve, we have 



= 2-Trt. 

Hence 4 (rjw TJ O)) = 27ri, 

and therefore i](f> rju> = \iri. 

This is the result as derived from (i), fig. 34, that is, when 91 [?-) is positive. 

\i/tU/ 

When (ii), fig. 34, is taken account of, the result is the same except 
that, when the circuit passes from z to z + 2&>, then to z + 2t + 2o> , 
then to z + 2&> and then to z , it passes in the negative direction round the 
parallelogram. The value of the integral along the path ABCDA is the 
same as before, viz., 4 (rjw rj a)) ; when the path is deformed into a small! 

rdz 

curve round the origin, the value of the integral is I taken negatively, an 

J ** 

therefore it is 2?ri : hence 

t]() rj (a = \Tri. 

Combining the results, we have 

rjay f]w = ^Tri, 
/ \ 
according: as 9t ( . ] is positive or negative. 

\0)lJ 

COROLLARY. If there be a change to any other fundamental parallelo 
gram, determined by 2H and 2O , where 

1 = pa) + qa) , 1 = p co + q a) , 
p, q, p , q being integers such that pq p q = 1, and if H, H denote 

C(ft ), then 

H = pr} + qrj , H = p rj + q f} ; 

therefore HW - H l = \-iri, 



according as the real part of T^ is positive or negative. 



130.] PRODUCT-FUNCTIONS 259 

130. It has been seen that (z) is pseudo-periodic ; there is also a pseudo- 
periodicity for o- (z), but of a different kind. We have 



that is, 

0-0 + 

and therefore a- (z + 2<w) = Ae^ z cr (z), 

where A is a constant. To determine A, we make z = &>, which is not a 
zero or an infinity of a (z) ; then, since a (z) is an odd function, we have 

so that o- + 2&>) = - e*> < z+<0 > a- (z). 

Hence o-(z + 4eo) = e*> (z+3ft)) <r(z + 2&>) 



and similarly a (z + 2mey) = ( l) m 

Proceeding in the same way from 



we find a~(z + 2m V) = (- l) m e*> < w/z + m 2w ) o- (z). 

Then a (z + 2ma> + 2m V) = (- l) m e 2 ^ (ww+^+V) Q- (^ + 2m / eo / ) 

== / _ J \m+m g sz (mij+m V) +2 ?m 2 <o+47]mmV+27) m 2co _. 



But lyct) r/o) = \iri, 

SO that g2mm (r|a> rj w) _ e mm iri _ /_ |\nj.m 

and therefore 



2m V) = (- l) w 

which is the law of change of a (z) for increase of z by integral multiples of 
the periods. 

Evidently <r(z) is not a periodic function, a result confirmatory of the 
statement in 122. But there is a pseudo-periodicity the characteristic of 
which is the reproduction, for an added period, of the function with an 
exponential factor the index being linear in the variable. This is another 
of the forms of pseudo-periodicity which will be considered in the next 
chapter. 

131. But though <r(z) is not periodic, we can by its means construct 
functions which are periodic in the pseudo-periods of a (z). 

By the result in the last section, we have 

<r (z a. + 2ma> + 2m &) ) cr (z a) + ,, 
<r(z-fi + 2mo> + 2m V) ~ <r(z ^~J3) & 

172 



260 DOUBLY-PERIODIC FUNCTIONS [131. 

and therefore, if <f> (z) denote 

a- (z cti) a (z 2 ) cr (z &) 



then $ (z + 2ra&> + 2m V) = e 2(m>} +m * > < 2 ^- 2 ^ 

so that $ (z) is doubly-periodic in 2 and 2&> provided 



Now the zeros of <f>(z), regarded as a product of o--functions, are a ls a 2 ,..., 
and the points homologous with them ; and the infinities are Pi, /3 2 , ... , ft n and i 
the points homologous with them. It may happen that the points a and ft j 
are not all in the parallelogram of reference ; if the irreducible points 
homologous with them be a 1} ..., a n and b lt ... , b n , then 

Sa r = ~S.br (mod. 2&>, 2co ), 

and the new points are the irreducible zeros and the irreducible infinities of 
<}>(z). This result, we know from Prop. III., 116, must be satisfied. 

It is naturally assumed that no one of the points a is the same as, or is 
homologous with, any one of the points ft : the order of the doubly-periodic 
function would otherwise be diminished by 1. 

If any a be repeated, then that point is a repeated zero of <j>(z); similarly- 
if any ft be repeated, then that point is a repeated infinity of < (z). In every, 
case, the sum of the irreducible zeros must be congruent with the sum of the 
irreducible infinities in order that the above expression for <j)(z) may be 
doubly-periodic. 

Conversely, if a doubly-periodic function < (z) be required with m assignedJ 
irreducible zeros a and m assigned irreducible infinities b, which are subject t 

to the congruence 

2a = 26 (mod. 2co, 2&> ), 

we first find points OL and ft homologous with a and with b respectively sucht 
that 



rru +U t 

Then the function 



a- (z - Pi) a(z ft m ) 

has the same zeros and the same infinities as </> (z), and is homoperiodic withi 
it ; and therefore, by 116, IV., 

o-(s-ai) o-^-otm) 

9 \ z > " ( _ o \ *(*ft V 



where A is a quantity independent of z. 

Ex. 1. Consider ft? (z). It has the origin for an infinity of the third degree and all thti 
remaining infinities are reducible to the origin ; and its three irreducible zeros are a, a/, a>" j 
Moreover, since <o"=a> + a>, we have w + w + w" congruent with but not equal to zerw 
We therefore choose other points so that the sum of the zeros may be actually the same, 



131.] EXAMPLES 261 

as the sum of the infinities, which is zero ; the simplest choice is to take <, &> , - ". 
Hence 



where A is a constant. To determine A, consider the expansions in the immediate 
vicinity of the origin ; then 

2 o- ( - co) <r ( - to ) v (a)") 

?"" ...... S 3 * ...... > 

sothat y^-g^^/rf^ tf. 

O- () or (eo ) o- (a ) O" 3 (2) 

Another method of arranging zeros, so that their sum is equal to that of the infinities, 
is to take w, , co" ; and then we should find 

dy M =2 
r W 

This result can, however, be deduced from the preceding form merely by changing the 
sign of z. 

Ex. 2. Consider the function 

. a- (u + v) a- (u v) 

*() 

where v is any quantity and A is independent of u. It is, qua function of u, doubly- 
periodic ; and it has u = as an infinity of the second degree, all the infinities being 
homologous with the origin. Hence the function is homoperiodic with g> (u) and it has 
the same infinities as $> (u) : thus the two are equivalent, so that 



where B and C are independent of u. The left-hand side vanishes if nv; hence 
(v), and therefore 



where A is a new quantity independent of u. To determine .4 we consider the 
expansions in the vicinity of u = ; we have 

A. <r(v)<r(-v) 



sothat 

and therefore cr- = 

o- 2 (%) o- 2 (v) 

a formula of very great importance. 

Ex. 3. Taking logarithmic derivatives with regard to u of the two sides of the last 
equation, we have 



and, similarly, taking them with regard to D, we have 



whence 



262 EXAMPLES [131. 

giving the special value of the left-hand side as ( 128) a doubly-periodic function. It is 
also the addition-theorem, so far as there is an addition-theorem, for the ^-function. 

Ex. 4. We can, by differentiation, at once deduce the addition-theorem for g) (u + v). 
Evidently 



which is only one of many forms : one of the most useful is 



which can be deduced from the preceding form. 

The result can be used to modify the expression for a general doubly-periodic function 
* (z) obtained in 128. We have 



Each derivative of f can be expressed either as an integral algebraical function of $ (z - a,.) 
or as the product of jjp (z a r ) by such a function ; and by the use of the addition-theorem 
these can be expressed in the form 



L > 
where L, M, N are rational integral algebraical functions of $(z). Hence the function 



can be expressed in the same form, the simplest case being when all its infinities are 
simple, and then 

4. (z) = C+ 2 A r {(e-ar) 



(*)- (Or) 



with the condition 2 A r = 0. 

r=l 

Ex. 5. The function $ (z) e 1 is an even function, doubly-periodic in 2 and 2o> and 
having 2 = for an infinity of the second degree ; it has only a single infinity of the second 
degi ee in a fundamental parallelogram. 

Again, z = &> is a zero of the function ; and, since ^X () = but $>" (o>) is not zero, it is a 
double zero of $ (z)-e l . All the zeros are therefore reducible to 2 = o> ; and the function 
has only a single zero of the second degree in a fundamental parallelogram. 

Taking then the parallelogram of reference so as to include the points = and 0=o>, 
we have 



where Q (z) has no zero and no infinity for points within the parallelogram. 

Again, for g> (z + o>) - e , the irreducible zero of the second degree within the parallelo- 



131.] OF DOUBLY-PERIODIC FUNCTIONS 263 



gram is given by S + <B = O>, that is, it is = 0; and the irreducible infinity of the second 
degree within the parallelogram is given by z + a = 0, that is, it is z = v. Hence we have 



where Q l (z) has no zero and no infinity for points within the parallelogram. 

Hence {> (z} - ej {%> (z + ) - ej m Q (z) Q 1 (z\ 

that is, it is a function which has no zero and no infinity for points within the 
parallelogram of reference. Being doubly-periodic, it therefore has no zero and no infinity 
anywhere in the plane ; it consequently is a constant, which is the value for any point. 
Taking the special value s = a>, we have jp( m ) = e s , and (jf>(a> + a>) = e 2 ; and therefore 

{#> (*) ~ e,} (V (* + )- e,} = (e 3 - *i) (e, - *i). 

Similarly {#> (z) - e 2 } {#> (z + ") - e 2 } = (e x - e 2 ) (e s - e 2 ), 

and {#> (2) - ^ {p (z + a) ) - e 3 } = (e 2 - e 3 } (^ - <? 3 ). 

It is possible to derive at once from these equations the values of the ^-function for 
the quarter-periods. 

Note. In the preceding chapter some theorems were given which indicated that 
functions, which are doubly-periodic in the same periods, can be expressed in terms of one 
another : in particular cases, care has occasionally to be exercised to be certain that the 
periods of the functions are the same, especially when transformations of the variables are 
effected. For instance, since g) (z) has the origin for an infinity and sn u has it for a zero, 
it is natural to express the one in terms of the other. Now $ (z) is an even function, and 
sn u is an odd function ; hence the relation to be obtained will be expected to be one 
between (z) and sn 2 w. But one of the periods of sn 2 u is only one-half of the correspond 
ing period of sn u ; and so the period-parallelogram is changed. The actual relation* is 

(P (z) - <? 3 = (<?!- e 3 ) sn- 2 , 
where u = (e l -e s f z and F = (<? 2 -e^)l(e l -e :i ). 

Again, with the ordinary notation of Jacobian elliptic functions, the periods of sn z are 
4 A" and 2iA", those of dn z are 2 A and 4i K , and those of en z are 4 A and 2A + 2iA . The 
squares of these three functions are homoperiodic in 2K and ZiK ; they are each of the 
second order, and they have the same infinities. Hence sn 2 z, en 2 z, dn 2 z are equivalent to 
one another ( 116, V.). 

But such cases belong to the detailed development of the theory of particular classes of 
functions, rather than to what are merely illustrations of the general propositions. 

132. As a last illustration giving properties of the functions just 
considered, the derivatives of an elliptic function with regard to the periods 
will be obtained. 

Let (/> (z) be any function, doubly-periodic in 2o> and 2&/ so that 

</> (z + 2m&> + 2m V) = </> (z), 

the coefficients in <f> implicitly involve &> and CD . Let <f> 1} < 2 , and </> respec 
tively denote 90/3t, 9</9o/, 9^/9^ ; then 

^ (z + 2771&) + 2m to ) + %m<j> (z + 2mo> + 2wV) = fa (z), 
fa (z + 2m&) + 2??iV) -1- 2m < (z + 2rao> + 2wV) = fa (z), 

$ (z + 2m&) + 2m V) = <f> (z). 

" Halphen, Fonctions Elliptiques, t. i, pp. 23 25. 



264 PERIOD-DERIVATIVES [132. 

Multiplying by &>, ro , z respectively and adding, we have 
&></>! (z + 2mo> + 2m V) + o> </> 2 (z + 2m< + 2mV) 

+ (z + 2mw + 2wV) </> + 2mo) 

= 0)0! (Z) + 0) (f). 2 (Z) + Z<f> (Z). 

Hence, if f(z} = mfa (z) + 6/0 2 (z) + z$ (z), 

then f(z) is a function doubly -periodic in the periods of (f>. 

Again, multiplying by rj, 77 , %(z), adding, and remembering that 
+ 2mm + 2raV) = ($) 



we have 

77$! (z + 2mw + 2m V) + 7/< 2 + 2mo + 2m V) 

+ (z + 2mm + 2m ay ) <f> (z + 2mw + 2m a> ) 

-ik<fi)+J+,(*) + f(*Wto 

Hence, if g (z) = yfa (z) + q fa (z) + (z) $ (z), 

then g(z) is a function doubly-periodic in the periods of <f>. 

In what precedes, the function <f>(z) is any function, doubly-periodic in 
2o>, 2&) ; one simple and useful case occurs when (z) is taken to be the 
function z. Now 



and fW-J-^ 

hence, in the vicinity of the origin, we have 

9 P >d@ d& 2 

(o ^- + 03 5*7 + jp-- as + even integral powers of z 1 
d(o d(o dz z- 

= -2^>, 

since both functions are doubly-periodic and the terms independent of z 
vanish for both functions. It is easy to see that this equation merely 
expresses the fact that <p, which is equal to 

l 



is homogeneous of degree 2 in z, &>, to . 
Similarly 

9|J> / ty ^d%> 22 

77 2+*) ~r-/ + b (- 2r ) y- = - ~i + j^ 9-i + even integral powers of z. 

But, in the vicinity of the origin, 

,5-7 = + YQ^ -I- even integral powers of 2, 



132.] OF WEIEHSTRASS S FUNCTION 265 

so that 

9P / d@ / \ dP 1 3 2 lP x r 

17 3^- + V g :S + f<) |^t i gji i** even mte g ral powers of z. 

The function on the left-hand side is doubly-periodic : it has no infinity 
at the origin and therefore none in the fundamental parallelogram ; it there 
fore has no infinities in the plane. It is thus constant and equal to its value 
anywhere, say at the origin. This value is ^g z> and therefore 



T/w s equation, when combined with 



, 

+ eo ; + z = - ty, 
dco da> oz 



j 

gives the value of ~- and -^ , . 
J dm 9&) 

The equations are identically satisfied. Equating the coefficients of z 2 in 
the expansions, which are valid in the vicinity of the origin, we have 



and equating the coefficients of ^ in the same expansions, we have 




Hence for any function u, which involves w and &/ and therefore implicitly 
involves g 2 and ^r 3 , we have 

du ,du 
w 5- + w , = 

aw a&) 

9w . , 3w 
17 a - +T; = - 
9&) 9&) 

Since ^) is such a function, we have 



f : * 

being ^/te equations which determine the derivatives of $ with regard to the 
invariants g., and g.^. 



266 EVEN [132. 

The latter equation, integrated twice, leads to 
9V da- 2 80- 1 



a differential equation satisfied by <r(z)*. 

133. The foregoing investigations give some of the properties of doubly- 
periodic functions of the second order, whether they be uneven and have two 
simple irreducible infinities, or even and have one double irreducible infinity. 

If a function U of the second order have a repeated infinity at z = y, then 
it is determined by an equation of the form 



or, taking U - (X + fi + v) = Q, the equation is 

Q = 4a 2 [(Q - e,) (Q - e,) (Q - $,)]*, 

where ^ + e 2 + e 3 = 0. Taking account of the infinities, we have 

Q=@(az- ay) ; 

and therefore U-(\ + /Ji. + v) = %> (az - ay) 

. . 1 (tp (az) + cp (ay)} 2 
= -Q (az) - <o (ay) + 

a x o x " 



by Ex. 4, p. 262. The right-hand side cannot be an odd function; hence 
an odd function of the second order cannot have a repeated infinity. Similarly, 
by taking reciprocals of the functions, it follows that an odd function of the 
second order cannot have a repeated zero. 

It thus appears that the investigations in 120, 121 are sufficient for the 
included range of properties of odd functions. We now proceed to obtain 
the general equations of even functions. Every such function can (by 118, 
XIII., Cor. I.) be expressed in the form |a#> (z) +b}+ {c#> (z) + d], and its 
equations could thence be deduced from those of p(z)\ but, partly for 
uniformity, we shall adopt the same method as in 120 for odd functions. 
And, as already stated (p. 251), the separate class of functions of the second 
order that are neither even nor odd, will not be discussed. 

134. Let, then, <j>(z) denote an even doubly-periodic function of the 
second order (it may be either of the first class or of the second class) and let 
2&), 2<w be its periods ; and denote 2&) + 2ft) by 2o>". Then 

since the function is even ; and since 

< (ft) + Z) = <f> ( &) z) 

= <f> (2&) &) z) 
= (j) (CD *), 

* For this and other deductions from these equations, see Frobenius und Stickelberger, Crelle, 
t. xcii, (1882), pp. 311327; Halphen, Traite des feme t ions elliptiqucs, t. i, (1886), chap. ix. ; 
and a memoir by the author, quoted on p. 254, note. 



134.] DOUBLY-PERIODIC FUNCTIONS 

it follows that < (&> + z) and, similarly, $ (&> + z) and (to" + z) are even 
functions. 

Now </> (w + a), an even function, has two irreducible infinities, and is 
periodic in 2&>, 2&/ ; also < (z), an even function, has two irreducible infinities 
and is periodic in 2&>, 2&/. There is therefore a relation between (z) and 
</> (w +z), which, by 118, Prop. XIII., Cor. I., is of the first degree in < (z) and 
of the first degree in <j) (&> + z) ; thus it must be included in 

B<f> (z) <j>(<o + z)-C<l> (z) -C <t>(a> + z)+A = 0. 

But < (z) is periodic in 2<w ; hence, on writing z + <w for z in the equation, it 

becomes 

B<f>(a>+z)<j>(z)-C<f>(co+z)-C <l>(z) + A=0; 

thus tf=C". 

If B be zero, then (7 may not be zero, for the relation cannot become 
evanescent : it is of the form 



A .............................. (1). 

If B be not zero, then the relation is 



Treating <f> (w + z) in the same way, we find that the relation between it 

and (f) (z) is 

F(j> (z) (f> (ay + z)-D(j> (z} -D(j>(a> + z) + E = 0, 

so that, if F be zero, the relation is of the form 

(*) + 0(a> + *) = J ........................... (I) , 

and, if F be not zero, the relation is of the form 



Four cases thus arise, viz., the coexistence of (1) with (1) , of (1) with (2) , 
of (2) with (1) , and of (2) with (2) . These will be taken in order. 

I. : the coexistence of (1) with (1) . From (1) we have 

<j> (a> + z} + (j> (&>" + z) = A , 

so that </> (z) + <f) (w + z) + (f) (w + z) + (<w" + z) = 2A . 

Similarly, from (1) , 



so that A = E , arid then 

(f)((0 + z)=(j)((o + Z\ 

whence <w ~ &> is a period, contrary to the initial hypothesis that 2&> and 2&> 
determine a fundamental parallelogram. Hence equations (1) and (1) cannot 
coexist. 



268 EVEN [134. 

II. : the coexistence of (1) with (2) . From (1) we have 
<(" + z} = A - <(&> + z) 



on substitution from (2) . From (2) we have 



cb (co + z) = -5*1) -- ( =r 
F<f) (CD + z) - D 

_ (A D -E)-D<j> (z) 
= A F - D - F(f> (z) 
on substitution from (1). The two values of < (&>" + z) must be the same, 

whence 

A F-D = D, 

which relation establishes the periodicity of </> (z) in 2ft)", when it is considered 
as given by either of the two expressions which have been obtained. We 

thus have 

A F=W- 
and then, by (1), we have 

<f>(z)-j+<l>( 
and, by (2) , we have 



If a new even function be introduced, doubly-periodic in the same periods 
having the same infinities and defined by the equation 

& 0) = </> 0) - J > 
the equations satisfied by fa (z) are 

fa(a> + z) + fa(z) = } 

fa (&) + z) fa (z) = constant] 

To the detailed properties of such functions we shall return later ; meanwhile 
it may be noticed that these equations are, in form, the same as those satisfied 
by an odd function of the second order. 

III. : the coexistence of (2) with (1) . This case is similar to II., with the 
result that, if an even function be introduced, doubly-periodic in the same 
periods having the same infinities and defined by the equation 

C 

fa (Z) = < (2) - -g , 

the equations satisfied by fa (z) are 

fa (&> + z) + fa (z) = } 

fa (&) + z) fa (z) = constant] 

It is, in fact, merely the previous case with the periods interchanged. 



134.] DOUBLY-PERIODIC FUNCTIONS 269 

IV. : the coexistence of (2) with (2) . From (2) we have 



_ (CD - AF) <ft (z) - (GE - AD) 
~ (BD - CF) <J> (z) - (BE - CD) 

on substitution from (2) . Similarly from (2) , after substitution from (2), we 
have 



~ 



The two values must be the same ; hence 

CD-AF=-(GD-BE\ 

which indeed is the condition that each of the expressions for <ft (&>" + z) 
should give a function periodic in 2&>". Thus 



One case may be at once considered and removed, viz. if C and D vanish 
together. Then since, by the hypothesis of the existence of (2) and of (2) , 
neither B nor F vanishes, we have 

A__E 

B~ F 

so that u + , = 



and then the relations are < (&> + z) + <f) (&> + z) = 0, 

or, what is the same thing, <ft (Y) + <ft (&>" + z) ] 

and </> (z) </>(&> + z) = constant j 

This case is substantially the same as that of II. and III., arising merely 

from a modification ( 109) of the fundamental parallelogram, into one whose 

sides are determined by 2&> and 2&>". 

Hence we may have (2) coexistent with (2) provided 

AF + BE=WD; 
C and D do not both vanish, and neither B nor F vanishes. 

IV. (1). Let neither C nor D vanish ; and for brevity write 

<f>((o + z)=<l> 1 , (f> (w" + z) = <o, </> (&) + z) = $3, (f) (z) = 0. 
Then the equations in IV. are 



Now a doubly-periodic function, with given zeros and given infinities, is 
determinate save as to an arbitrary constant factor. We therefore introduce 
an arbitrary factor X, so that 

<=Xi/r, 

G D 

and then taking = CI = Ca 



270 EVEN [134. 


we have (^ - Cl ) (fa - Cj ) = d 2 - -^ , 

ET 



The arbitrary quantity A, is at our disposal : we introduce a new quantity c 2 , 
defined by the equation 

A 

Tt-. o Ci (C 2 + 3) C 2 C 3 , 

and therefore at our disposal. But since 

AF + BE=2CD, 

A E .CD 

we have ^ + ^ = 2 ^ ^ = 2 Cl c 3j 

ri 

and therefore ^-- 2 = c 3 (Cj + c 2 ) - 0^2 . 

Hence the foregoing equations are 

- d) = (Cj - C 2 ) (d - C 3 ), 

- C 3 ) = (C 3 - d) (C 3 - C 2 ). 

The equation for ^> 2 , that is <f>((o" + z), is 

_Lcf)-M 



where L= CD - BE = AF - CD, M=AD-CE, N=CF-BD, 

so that ^ + 5M" = 2CL. 

As before, one particular case may be considered and removed. If N be 

zero, so that 

C_D_ 

B~F~ a 

AE CD ,. 

say, and B + F = RF = 

then we find $ + ^> 2 = ^>i + < 3 = 2, 

or taking a function ^ = a, 

the equation becomes % (^ + % C^" + ^ = 0. 
The other equations then become 



and therefore they are similar to those in Cases II. and III. 
If N be not zero, then it is easy to shew that 
N=BF\(c 1 -c 3 ) > 



M = BF\ 3 (d - c 3 ) (c.,C! + c,c 3 - dc 3 ) ; 



134.] DOUBLY-PERIODIC FUNCTIONS 271 

and then the equation connecting and 2 changes to 



s - Ca) = (C a - C x ) (Ca - C s ) 

which, with (^ d) ("^i d) = (d c 2 ) (d c 3 ) 

( r ^ 3 / \ i 3 ^ 3 / == V^ 3 ^" ^ 3 ^ 2 

are relations between ty, ^r l} -^ 2 , ty. 3 , where the quantity c 2 is at our disposal. 

IV. (2). These equations have been obtained on the supposition that 
neither G nor D is zero. If either vanish, let it be C: then D docs not 
vanish ; and the equations can be expressed in the form 

E 



D\ 

J 

E\ E(D*-EF) 



We therefore obtain the following theorem : 

If (f> be an even function doubly-periodic in 2&> and 2&> and of the second 
order, and if all functions equivalent to <J> in the form R<f> + 8 (where R and 
S are constants) be regarded as the same as 0, then either the function satisfies 
the system of equations 



00) 0O" 
where H is a constant ; or it satisfies the system of equations 

{0 0) - d} {0 (ft) +Z)- d] = (Ci - C 2 ) (d - C 3 ) 
{00)-C 3 }{0(>/ +^)-C 3 }=(Cs-C 1 )(Cs-Ca) 
{0 0) - C 2 } (0 ((,)" + Z)- C 2 } = (C 2 ~ d) (C 2 - Cs) 

where of the three constants c lt c 2 , c s one can be arbitrarily assigned. 
We shall now very briefly consider these in turn. 

135. So far as concerns the former class of equations satisfied by an even 
doubly-periodic function, viz., 



we proceed initially as in ( 120) the case of an odd function. We have the 
further equations 

00) = 0(-4 

(ft) + Z) (ft) Z), (a/ + Z) = (ft) Z). 

* The systems obtained by the interchange of w, w , w" among one another in the equations 
are not substantially distinct from the form adopted for the system I. ; the apparent difference 
can be removed by an appropriate corresponding interchange of the periods. 



272 EVEN DOUBLY-PERIODIC FUNCTIONS [135. 

Taking z = ^w, the first gives 



so that ^&> is either a zero or an infinity. 
If \<> be a zero, then 

(f> (f to) = $ (< + ^ft)) = <f> (^) by the first equation 

= 0, 
so that ^&> and f&> are zeros. And then, by the second equation, 

&) + ^<w, &) 4- f a) 
are infinities. 

If \w be an infinity, then in the same way |w is also an infinity ; and 
then a) + \w, &> + f &) are zeros. Since these amount merely to interchanging 
zeros and infinities, which is the same functionally as taking the reciprocal of 
the function, we may choose either arrangement. We shall take that which 
gives ^0), f &> as the zeros ; and &> 4- ^&>, &/ + f &> as the infinities. 

The function <j> is evidently of the second class, in that it has two distinct 
simple irreducible infinities. 

Because &> + |&), &> + f &> are the irreducible infinities of </> (z), the four 
zeros of $ (z} are, by 117, the irreducible points homologous with &>", 
&)" + &>, &>" + a) , a)" + &)", that is, the irreducible zeros of (f) (z) are 0, &>, &> , &>". 
Moreover 



by the first of the equations of the system ; hence the relation between (f> ( 
and ((> (z) is 

#* (z} = A{<t>(z)-$ (())} {</> (z) - (/> ()} |0 (*) - (/> (ft) )} {(/> (*) - </> (")} 

= A [p (0) - p (z)}{p (ft) ) - ^ (*)}. 
Since the origin is neither a zero nor an infinity of < (^), let 



so that </>j (0) is unity and 0/ (0) is zero ; then 

^(*)X{l-^(*)}{^-^(f)) 

the differential equation determining fa (z). 

The character of the function depends upon the value of p and the 
constant of integration. The function may be compared with en u, by taking 

2ft), 2&/ = 4>K, 2K + 2iK ; and with * , by taking 2ft), 2ft) = 2K, MK , 

dn u 

which ( 131, note) are the periods of these (even) Jacobian elliptic functions. 
We may deal even more briefly with the even, function characterised by 
the second class of equations in 134. One of the quantities c 1} c 2 , c 3 being 
at our disposal, we choose it so that 

Ci + c 2 + c 3 = ; 

and then the analogy with the equations of Weierstrass s ^-function is 
complete (see 133). 



CHAPTER XII. 

PSEUDO-PERIODIC FUNCTIONS. 

136. MOST of the functions in the last two Chapters are of the type 
called doubly-periodic, that is, they are reproduced when their arguments are 
increased by integral multiples of two distinct periods. But, in 127, 130, 
functions of only a pseudo-periodic type have arisen : thus the ^-function 
satisfies the equation 

m2&> + m 2&> ) = ( + m2i) + m 2v , 
,nd the cr-function the equation 

m i (mr,+m r, ) (z+wuo+m oi 1 ) 



These are instances of the most important classes: and the distinction 
between the two can be made even less by considering the function 
e^ (z} ^(z), when we have 

(z + ra2&> + m 2&> ) = e- mr > e" * % (z). 

In the case of the ^-function an increase of the argument by a period leads 
to the reproduction of the function multiplied by an exponential factor that 
is constant, and in the case of the <r-function a similar change of the 
argument leads to the reproduction of the function multiplied by an 
exponential factor having its index of the form az + b. 

Hence, when an argument is subject to periodic increase, there are three 
simple classes of functions of that argument. 

First, if a function f(z) satisfy the equations 

/(* + 2fi>) =/(*), /(* + 2 ) =/(*), 

it is strictly periodic : it is sometimes called a doubly-periodic function of the 
first kind. The general properties of such functions have already been 
considered. 

Secondly, if a function F(z) satisfy the equations 

F (z + 2&>) = pF (z), F (z + 2&/) - pfF (z), 
F - 18 



274 THREE KINDS [136. 

where /u, and fjf are constants, it is pseudo-periodic : it is called a doubly- 
periodic function of the second kind. The first derivative of the logarithm 
of such a function is a doubly-periodic function of the first kind. 

Thirdly, if a function <f> (z) satisfy the equations 

<j>(z + 2o)) = e az+b <j> (z\ <f>(z + 2ft) ) = e a z+v (j> (z), 

where a, b, a , b are constants, it is pseudo-periodic : it is called a doubly- 
periodic function of the third kind. The second derivative of the logarithm 
of such a function is a doubly-periodic function of the first kind. 

The equations of definition for functions of the third kind can be 
modified. We have 

. <f> (Z + 2ft) + 2ft) ) = e (2+2<o )+6+a z+6 < ( z ) 
ga (2+2o>) +b +az+b J, f z \ 

whence a oo am = nnri, 

where ra is an integer. Let a new function E (z) be introduced, defined by 

the equation 

()*"+* t(*)i 

then X and /A can be chosen so that E (z} satisfies the equations 

E(z + 2a)) = E (z\ E(z+ 2ft) ) = e Az+ E (z\ 
From the last equations, we have 

E (z + 2&) + 2ft) ) = eA(*+**+B E ^ 

= e Az+s E (z), 
so that 2Aa) is an integral multiple of 2? . 

Also we have E(z + 2o>) = e*(*-* +^+a) <j>( z + 2o>) 



so that 4X&) + a = 0, 

and 4A,ftr + 2/A&) +6 = (mod. 

Similarly, E (z + 2ft) ) = e^+wj +^+w, ^ + 2ft) ) 



so that 4Xo) + a = A, 

and 4W 2 + 2/^co + 6 = B (mod. 27ri). 

From the two equations, which involve X and not //,, we have 

Aco = a o) aw 



agreeing with the result with 2 A co is an integral multiple of Ziri. 

And from the two equations, which involve /j,, we have, on the elimination 
of /j, and on substitution for X and A, 

b co 6ft) a&) (ft) &)) = 5ft) (mod. 2-Tn ). 



136.] OF DOUBLY-PERIODIC FUNCTIONS 275 

If A be zero, then E(z) is a doubly-periodic function of the first kind 
when e B is unity, and it is a doubly-periodic function of the second kind 
when e B is not unity. Hence A, and therefore m, may be assumed to be 
different from zero for functions of the third kind. Take a new function 
3?z such that 



mm 
then <l> (z) satisfies the equations 



4) (z + 2&)) = <I> (z\ <&(z + 2o) ) = e w 3>(z) 

* / \ / \ / \ /t 

which will be taken as the canonical equations defining a doubly -periodic 
function of the third kind. 

Ex. Obtain the values of X, p, A, B for the Weierstrassian function ir(z). 

We proceed to obtain some properties of these two classes of functions 
which, for brevity, will be called secondary-periodic functions and tertiary- 
periodic functions respectively. 

Doubly-Periodic Functions of the Second Kind. 

For the secondary-periodic functions the chief sources of information are 

Hermite, Comptes Rendus, t. liii, (1861), pp. 214228, ib., t. Iv, (1862), pp. 1118, 
85 91 ; Sur quelques applications des fonctions elliptiques, I in, separate 
reprint (1885) from Comptes Rendus; "Note sur la theorie des fonctions ellip 
tiques" in Lacroix, vol. ii, (6th edition, 1885), pp. 484491; Cours d Analyse, 
(4 me ed.), pp. 227234. 

Mittag-Leffler, Comptes Rendus, t. xc, (1880), pp. 177 180. 

.Frobenius, Crelle, t. xciii, (1882), pp. 53 68. 

Brioschi, Comptes Rendus, t. xcii, (1881), pp. 325328. 

Halphen, Traite des fonctions elliptiques, t. i, pp. 225 238, 411 426, 438 442, 463. 

137. In the case of the periodic functions of the first kind it was proved 
that they can be expressed by means of functions of the second order in the 
same period these being the simplest of such functions. It will now be 
proved that a similar result holds for secondary- periodic functions, defined by 
the equations 



Take a function Q (z} = 



a (z) a- (a) 

then we have G(z+2a>) = <r (* + g 

a (a) a (z + 2w) 



arid G(z+ 2&/) = e V+2W Q. (^). 

The quantities a and X being unrestricted, we choose them so that 

_ g2rja+2A<o __ g2T) a+2A(o 

and then G (z), a known function, satisfies the same equation as F (z). 

182 



276 PSEUDO-PERIODIC FUNCTIONS [137. 

Let u denote a quantity independent of z, and consider the function 

f( Z ) = F(z)G(u-z}. 
We have f(z + 2o>) = F(z + 2o>) G (u- z - 2w) 



=/(*) ; 

and similarly f(z + 2<o ) =f(z), 

so that/(X) is a doubly-periodic function of the first kind with 2 and 2o> 

for its periods. 

The sum of the residues of f(z) is therefore zero. To express this sum, 
we must obtain the fractional part of the function for expansion in the 
vicinity of each of the (accidental) singularities of f(z), that lie within the 
parallelogram of periods. The singularities of/ (2) are those of G (u z) and 
those of F(z). 

Choosing the parallelogram of reference so that it may contain u, we have 
z = u as the only singularity of G (u z) and it is of the first order, so that, 
since 

$() =+ positive integral powers of f 
in the vicinity of = 0, we have, in the vicinity of u, 
f(z) = {F (u) + positive integral powers of u z} \ -4- positive powers I 

= -- + positive integral powers of z u ; 

hence the residue of/(Y) for u is F(u}. 

Let z = c be a pole of F (z) in the parallelogram of order n + 1 ; and, in 
the vicinity of c, let 

(?! _ cf / 1 \ d n ( 1 \ 

F(z) = ^ c + G ^ Z (jr^J + + C n+i fan (zITc) + P sltlve integral powers. 

Then in that vicinity 



and therefore the coefficient of - in the expansion of f(z) for points in the 

Z ~~ 

vicinity of c is 



which is therefore the residue off(z) for c. 

This being the form of the residue of f(z) for each of the poles of F (z), 
then, since the sum of the residues is zero, we have 



137.] OF THE SECOND KIND 277 

or, changing the variable, 



,. .. n+l n - 

where the summation extends over all the poles of F(z) within that parallelo 
gram of periods in which z lies. This result is due to Hermite. 

138. It has been assumed that a and \, parameters in 0, are determinate, 
an assumption that requires /j, and ^ to be general constants : their values 
are given by 

yd 4- &>X = | log fjb, r) a + &/X \ log //, 

and, therefore, since ijca rfca = ^ITT, we have 

+ ITTCL = w log /JL co log //) 

+ iir\ = V) log /i + 77 log /z j 

Now X may vanish without rendering G (z) a null function. If a vanish (or, 
what is the same thing, be an integral combination of the periods), then G (z) 
is an exponential function multiplied by an infinite constant when X does not 
vanish, and it ceases to be a function when X does vanish. These cases must 
be taken separately. 

First, let a and X vanish* ; then both //, and /// are unity, the function F 
is doubly-periodic of the first kind ; but the expression for j^is not determinate, 
owing to the form of G. To render it determinate, consider X as zero and a 
as infinitesimal, to be made zero ultimately. Then 

,, o-(z) + aa (z) + ... .^ 

(*(z) = - - ~ (1 + positive integral powers of a) 

= - + (z) + positive powers of a. 
a 

Since a is infinitesimal, /JL and /j, are very nearly unity. When the 
function F is given, the coefficients C 1} <7 2 , ... may be affected by a, so that 
for any one we have 

Ck b k + ay k + higher powers of a, 

where y h is finite ; and b k is the actual value for the function which is strictly 
of the first kind, so that 

Sk-O, 

the summation being extended over the poles of the function. Then retaining 
only a" 1 and a, we have 



This case is discussed by Hermite (I.e., p. 275). 



278 MITTAG-LEFFLER S THEOREM [138. 

where C , equal to 71, is a constant and the term in - vanishes. This expres- 

CL 

sion, with the condition S^ = 0, is the value of F (u) or, changing the variables, 
we have 



with the condition S&i = 0, a result agreeing with the one formerly ( 128) 
obtained. 

When F is not given, but only its infinities are assigned arbitrarily, then 
SO = because F is to be a doubly-periodic function of the first kind ; the 

term - "C vanishes, and we have the same expression for F(z) as before. 
Secondly, let a vanish* but not \, so that ^ and // have the forms 



We take a function g (z) = 

then g(z- 2o>) = ^ e^ (z - 2eo ) 



and g(z-2a> ) = p - 1 {g (z} - 2?/ e^} . 

Introducing a new function H (z) defined by the equation 



we have H (z + 2t) = H (z) - 2ije A <-* F (z), 

and H (z + 2o> ) = H (z) - 27?V< M -*> F(z). 

Consider a parallelogram of periods which contains the point u ; then, if be 

the sum of the residues of H (z) for poles in this parallelogram, we have 



the integral being taken positively round the parallelogram. But, by 116, 
Prop. II. Cor., this integral is 



f e-*(p+*-) F (p + 2tot) dt - 0/77 f e-^ + 
Jo Jo 



where p is the corner of the parallelogram and each integral is taken for real 
values of t from to 1. Each of the integrals is a constant, so far as concerns 
u ; and therefore we may take 

= -Ae^ u , 

the quantity inside the above bracket being denoted by \i-rrA. 

The residue of H (z) for z = u, arising from the simple pole of g (u z), is 
-F(u) as in 137. 

If z = c be an accidental singularity of F (z) of order n+1, so that, in the 
vicinity of z = c, 

F(.) = C, + 0. A- + . . . + BU i- + P (, - c), 



This is discussed by Mittag-Leffler, (I.e., p. 275). 



138.] ON SECONDARY FUNCTIONS 279 

then the residue of H (z) for z = c is 

d d n 



and similarly for all the other accidental singularities of F (z}. Hence 



F(z) = A** + 

where the summation extends over all the accidental singularities of F (z) in a 
parallelogram of periods which contains z, and y (z) is the function e xz %(z}. 
This result is due to Mittag-Leffler. 

Since /* = e 2 *" and 

g (z - c + 2&>) = fig (z - c) + 
we have 



and therefore 2 (G l + C. 2 \ 4- . . . + G n+l \ n ) e~^ = 0, 

the summation extending over all the accidental singularities of F(z). The 

same equation can be derived through ^F(z) = F(z + 2&> ). 

Again 2(7 : is the sum of the residues in a parallelogram of periods, and 
therefore 



the integral being taken positively round it. If p be one corner, the integral 

n 

F (p + 2co t) dt, 

Jo 



IS 

/i n 



o 
, each integral being for real variables of t. 

Hermite s special form can be derived from Mittag-Leffler s by making \ 
vanish. 

Note. Both Hermite and Mittag-Leffler, in their investigations, have 
used the notation of the Jacobian theory of elliptic functions, instead of 
dealing with general periodic functions. The forms of their results are as 
follows, using as far as possible the notation of the preceding articles. 

I. When the function is denned by the equations 

F (z + 2K) = ^F (z), F(z+ 2iK ) = ^F (z), 

then F(z) = 



280 INFINITIES AND ZEROS [138. 

(the symbol H denoting the Jacobian .ff-function), and the constants <w and X 
are determined by the equations 



II. If both X and to be zero, so that F(z) is a doubly-periodic function 
of the first kind, then 



with the condition 5$i = 0. 

III. If W be zero, but not X, then 



... 
where g (z} = --& V, 



the constants being subject to the condition 

2 (G, + C,\ + . . . + G n+1 X")e- Ac = 0, 

and the summations extending to all the accidental singularities of F(z) in a 
parallelogram of periods containing the variable z. 

139. Reverting now to the function F(z) we have G (z), defined as 



a (z) a (a) 

when a and X are properly determined, satisfying the equations 
G(z + 2a>) = ftG (z), + 2 & /) = yu/0). 

Hence H (z) = F(z)/G (z) is a doubly-periodic function of the first kind ; and 
therefore the number of its irreducible zeros is equal to the number of its 
irreducible infinities, and their sums (proper account being taken of multipli 
city) are congruent to one another with moduli 2 and 2&> . 

Let Ci, c 2 ,..., c m be the set of infinities of F (z) in the parallelogram of 
periods containing the point z ; and let y : , . . . , 7^ be the set of zeros of F (z) in 
the same parallelogram, an infinity of order n or a zero of order n occurring 
n times in the respective sets. The only zero of (z) in the parallelogram is 
congruent with a, and its only infinity is congruent with 0, each being 
simple. Hence the m+l irreducible infinities of H (z) are congruent with 

a, GI, GZ, . . . , c m , 
and its /* + 1 irreducible zeros are congruent with 

0, 71, 7s> >%*; 
and therefore m + 1 = p, + 1, 



139.] OF SECONDARY FUNCTIONS 281 

From the first it follows* that the number of infinities of a doubly-periodic 
function of the second kind in a parallelogram of periods is equal to the number 
of its zeros, and that the excess of the sum of the former over the sum of the 
latter is congruent with 

, (> i w i , 
+ , log it -. log u, 

- \7Tl TTl 6 

/ 

the sign being the same as that of 9t ( 

\10) 

The result just obtained renders it possible to derive another expression 
for F (z), substantially due to Hermite. Consider a function 

F ( Z ) = -Q-7i) 0-0-72).. -0-0-7) ePZ 

(T(z-c 1 )(r(z-c 2 )...ar(z-c m ) 

where p is a constant. Evidently F 1 (z) has the same zeros and the same 
infinities, each in the same degree, as F (z). Moreover 

F, (Z + 2ft)) = F l (Z) e 2,(2 C -2y) + 2p Wj 
F 1 (Z + 2ft) ) = F! (Z) e 2V(2e-2y)+2 P , t 

If, then, we choose points c and 7, such that 

Sc 7 = a, 
and we take p = \ where a and X are the constants of G (z), then 

F, (z + 2co) = ^ (z), F, (z + 2ft> ) = n Fj. (z). 

The function F l (z)/F(z) is a doubly-periodic function of the first kind and by 
the construction of F l (z) it has no zeros and no infinities in the finite part of 
the plane: it is therefore a constant. Hence 

F(z] = A gfr- ftM*- /)* (**) ^ 

a(z- c,) a- (z - C 2 ). . .o- (z - C m ) 

where Sc 7 = a, and a and A, are determined as for the function G (z}. 

140. One of the most important applications of secondary doubly-periodic 
functions is that which leads to the solution of Lame s equation in the cases 
when it can be integrated by means of uniform functions. This equation is 
subsidiary to the solution of the general equation, characteristic of the 
potential of an attracting mass at a point in free space; and it can be 
expressed either in the form 

jY = (Ak 2 sn 2 z + B) w, 
or in the form - 2 - = (A@ (z) + B} w, 

* Frobenius, Crelle, xciii, pp. 55 68, a memoir which contains developments of the properties 
of the function G (z). The result appears to have been noticed first by Brioschi, (Comptes Ilendus, 
t. xcii, p. 325), in discussing a more limited form. 



282 LAMP S [140. 

according to the class of elliptic functions used. In order that the integral 
may be uniform, the constant A must be n (n -f 1), where n is a positive 
integer ; this value of A, moreover, is the value that occurs most naturally in 
the derivation of the equation. The constant B can be taken arbitrarily. 

The foregoing equation is one of a class, the properties of which have 
been established* by Picard, Floquet, and others. Without entering into 
their discussion, the following will suffice to connect them with the secondary 
periodic function. 

Let two independent special solutions be g (z) and h (z), uniform functions 
of z ; every solution is of the form ag (z} + /3h (z}, where a and /3 are constants. 
The equation is unaltered when z + 2w is substituted for z ; hence g {z + 2&>) 
and h (z + 2&>) are solutions, so that we must have 

g (z + 2w) = Ag (z} + Bh (z}, h(z + 2o>) = Cg (z) + Dh (z\ 

where, as the functions are determinate, A, B, C, D are determinate constants, 
such that AD BC is different from zero. 

Similarly, we obtain equations of the form 

g (z + 2co ) = A g (z) + B h (z\ h(z + 2co ) = C g (z} + D h (z}. 
Using both equations to obtain g (z + 2o> + 2&/) in the same form, we have 

BC = B C, AB + BD = A B + B D ; 
and similarly, for h (z + 2w + 20) ), we have 



C G A-D A -U 

therefore -~ = - = o, ~ = = e. 

x> -D n n 

Let a solution F (z} = ag (z) + bh (z) 

be chosen, so as to give 



if possible. The conditions for the first are 



a b 

so that a/b (= ) must satisfy the equation 

and the conditions for the second are 

aA + bC f aB + bD 



* Picard, Comptes Rendus, t. xc, (1880), pp. 128131, 293295; Crelle, t. xc, (1880), pp. 
281302. 

Floquet, Comptes Rendus, t. xcviii, (1884), pp. 82 85 ; Ann. de VEc. Norm. Sup., 3 mc Ser., 
t. i, (1884), pp. 181238. 



140.] DIFFERENTIAL EQUATION 283 

so that must satisfy the equation 

A -D =^B ~~. 
These two equations are the same, being 

p.-g-ft*a 

Let j and 2 be the roots of this equation which, in general, are unequal ; and 
let fa, fa and fa, fa. be the corresponding values of /z, //. Then two functions, 
say FI (z) and F^ (z), are determined : they are independent of one another, so 
therefore are g (z) and h (z) ; and therefore every solution can be expressed in 
terms of them. Hence a linear differential equation of the second order, having 
coefficients that are doubly-periodic functions of the first kind, can generally be 
integrated by means of doubly -periodic functions of the second kind. 

It therefore follows that Lame s equation, which will be taken in the form 



can be integrated by means of secondary doubly-periodic functions. 

141. Let z = c be an accidental singularity of w of order m ; then, for 
points z in the immediate vicinity of c, we have 



and therefore 



2mp 

~ z- c + P SltlVe P wers f * - 



Since this is equal to n (n + 1) @ (z) + B 

it follows that c must be congruent to zero and that m, a positive integer, 
must be n. Moreover, p = 0. Hence the accidental singularities of w are 
congruent to zero, and each is of order n. 

The secondary periodic function, which has no accidental singularities 
except those of order n congruent to z = 0, has n irreducible zeros. Let them 
be a lt a 2 ,..., a n ; then the form of the function is 



Hence 1 *? = ,-? + 



or, taking p = - ^(a r ), we have 



and therefore i *? - 1 (*?) . n( > (,) - X f> ( + , 
19 O^ W 2 \dzj * v y ri 



284 INTEGRATION 

But, by Ex. 3, 131, we have 



[141. 



4 r=1 > (a r ) - p (z) 



, 



by Ex. 4, 131. Thus 



W 

Now 



r=l S =l 



. 

g> (a.) - 



g> (a r ) - g) () g> (a,) - g> (^) 

4^? 3 (^r) - ^ 2 ip Q) - #, + %> (a*) & (a,) 



where 



g> (a r ) - > (a.) 
Let the constants a be such that 



(O - > (a 2 ) 






+ 



-H...-0 



/i equations of which only n 1 are independent, because the sum of the n 
left-hand sides vanishes. Then iu the double summation the coefficient of 

i f .1 f u #> ( a r) & (z) . 
each of the tractions * ),- V\ is zero ; and so 



and therefore -^-, = w (w + 1) p (z) + (2n 1) 2 ^> (a,.). 

/ IU GLZ" T=l 

Hence it follows that 

_<T(z + aJ <T(z + a 2 )...<r(z + a n ) - z ?J("r) 
a n (z} 

satisfies Lame s equation, provided the n constants a be determined by the 
preceding equations and by the relation 

B = (2n-l) I pfa). 



141.] OF LAMP S EQUATION 285 

Evidently the equation is unaltered when z is substituted for z ; and 
therefore 



is another solution. Every solution is of the form 

MF(z} + NF(-z), 
where M and N are arbitrary constants. 

COEOLLARY. The simplest cases are when n = l and n = 2. 

When n = 1, the equation is 



j-r- + B : 

w dz z 

there is only a single constant a determined by the single equation 

B = p (a), 
and the general solution is 

,, a (2 + a) ... , r a(z a] ... , 
w = M ^- 7 -^- / e~ 2 (a) + N - ---- e s ^ a > 
o- (z} a (z) 

When n = 2, the equation is 



-J-. = 6(0 (z} + B. 

w dz* 

The general solution is 



^ 



where a and b are determined by the conditions 



Rejecting the solution a+b = 0, we have a and b determined by the equations 
p (a) 



For a full discussion of Lame s equation and for references to the original sources of 
information, see Halphen, Traite des fonctions elliptiques, t. ii, chap, xn., in particular, 
pp. 495 et seq. 

Ex. When Lamp s equation has the form 

1 d?w 
- -T-5 =n (n + 1) 2 sn 2 - h. 

w dz 2 ^ 

obtain the solution for w = l, in terms of the Jacobian Theta-Functions, 



where co is determined by the equation dn 2 o> = A-F ; and discuss in particular the solution 
when h has the values l+ 2 , 1, 2 . 

Obtain the solution for = 2 in the form 



i +B - fe^) e - K& .1 

J SI e() j 



286 PSEUDO-PERIODIC FUNCTIONS [141. 

where X and w are given by the equations 

(2P sn 2 a - 1 - F) (2F sn 2 a - 1) (2 sn 2 a - 1) 
3Fsn 4 a-2(l+ 2 )sn 2 a + l ~ 



and a is derived from h by the relation 



Deduce the three solutions that occur when X is zero, and the two solutions that occur 
when X is infinite. (Hermite.) 

Doubly-Periodic Functions of the Third Kind. 

142. The equations characteristic of a doubly-periodic function <I> (z) of 
the third kind are 

= <(, <&(z + 2a) ) = e~ ~ Z Q(z), 



where m is an integer different from zero. 

Obviously the number of zeros in a parallelogram is a constant, as well as 
the number of infinities. Let a parallelogram, chosen so that its sides 
contain no zero and no infinity of <& (z}, have p, p + 2<w, p + 2&> for three 
of its angular points; and let a 1} a 2 , . .., a { be the zeros and c l5 ..., c m be the 
infinities, multiplicity of order being represented by repetitions. Then using 

"^ (z) to denote , (log < (z)}, we have, as the equations characteristic of 



* 



and for points in the parallelogram 



where -ff (^) has no infinity within the parallelogram. Hence 



the integral being taken round the parallelogram : by using the Corollary to 
Prop. II. in 116, we have 

27ri (I - n) - - \ - \^L\ dz = 
Jp \ &> / 

so that I = n + m : 

or the algebraical excess of the number of irreducible zeros over the number of 

irreducible infinities is equal to in. 

z 
Again, since = 1 + 



z /A z p, 

a c 

we have 2 2 h I n = z"^ (z) zH (z), 

z a z c 

and therefore 2-Tn (Sa 2c) = jz*\? (z) dz, 



142.] OF THE THIRD KIND 287 

the integral being taken round the parallelogram. As before, this gives 

rp+2<a rp+2<a < vnTri "I 

2 (2a - 2c) = 2ft)^ (z) dz - MV (z) - - (z + 2ft/) dz. 

Jp Jp ( ft) ) 

The former integral is 

rp+* (g) 

, v x dz 
(*) 

miri 



for the side of the parallelogram contains* no zero and no infinity 
The latter integral, with its own sign, is 



<P(Z) ft) 

= + {O + 2 + 2ft> ) 2 - (p + 2ft/) 2 } 

= 2TO7T* (p + ft) + 2ft) ). 

Hence 2a Sc = m (&) + 2&/), 

giving the excess of the sum of the zeros over the sum of the infinities in any 
parallelogram chosen so as to contain the variable z and to have no one of its 
sides passing through a zero or an infinity of the function. 

These will be taken as the irreducible zeros and the irreducible infinities : 
all others are congruent with them. 

All these results are obtained through the theorem II. of 116, which 
assumes that the argument of <y is greater than the argument of &) or, what 
is the equivalent assumption ( 129), that 

rjco w co = ^iri. 

143. Taking the function, naturally suggested for the present class by 
the corresponding function for the former class, we introduce a function 



a(z- d) <r(z- C 2 ). ..<r(z C n ) 

where the a s and the c s are connected by the relations 
Sa Sc = m (&) + 2&> ), ln = m. 

Then (f>(z) satisfies the equations characteristic of doubly-periodic functions 
of the third kind, if 

= 4Xo) + 2ra77, 
k . 27rt = 4X&) 2 + 2m?/ft) + 2/ift) + miri Zmrj (&> + 2ft) ) ; 




miri 2mrj (&> + 2ft) ), 

* Both in this integral and in the next, which contain parts of the form I there is, as in 

J w 

Prop. VII., 116, properly an additive term of the form 2iciri, where K is an integer ; but, as there, 
both terms can be removed by modification of the position of the parallelogram, and this modifi 
cation is supposed, in the proof, to have been made. 



288 TERTIARY FUNCTIONS [143. 

k and k being disposable integers. These are uniquely satisfied by taking 



with A; = 0, k = m. 

Assuming the last two, the values of X and /JL are thus obtained so as to make 
<fr (z) a doubly-periodic function of the third kind. 

Now let Oj, ..., di be chosen as the irreducible zeros of <l> (z) and Ci, ..., c n 
as the irreducible infinities of <E> (2), which is possible owing to the conditions 
to which they were subjected. Then <3> (z)/<j> (z) is a doubly-periodic function 
of the first kind; it has no zeros and no infinities in the parallelogram of- 
periods and therefore none in the whole plane ; it is therefore a constant, so 
that 

3> (z) = Ae"** " IZ * + ^ - + (l|+8 ) } ** <r(*-gi)-(*-q)-. *(*-<*!) 

tr(z- d) <r(z- c. 2 )...o- (z - c n ) 

a representation of <3> (z) in terms of known quantities. 

Ex. Had the representation been effected by means of the Jacobian Theta-Functions 
which would replace a (z) by H(z), then the term in z 1 in the exponential would be absent. 

144. No limitation on the integral value of m, except that it must not 
vanish, has been made : and the form just obtained holds for all values. 
Equivalent expressions in the form of sums of functions can be constructed : 
but there is then a difference between the cases of m positive and m 
negative. 

If m be positive, being the excess of the number of irreducible zeros over 
the number of irreducible infinities, the function is said to be of positive class 
m ; it is evident that there are suitable functions without any irreducible 
infinities they are integral functions. 

When m is negative (= n), the function is said to be of negative class n ; 
but there are no corresponding integral functions. 

145. First, let m be positive. 

i. If the function have no accidental singularities, it can be expressed in 
the form 

A e**+i* a-(z a 1 )a-(z a a )...<r(z a m ), 

with appropriate values of X and //.. 

ii. If the function have n irreducible accidental singularities, then it has 
m + n irreducible zeros. We proceed to shew that the function can be 
expressed by means of similar functions of positive class m, with a single 
accidental singularity. 



145.] OF POSITIVE CLASS 289 

Using X and /j, to denote 



, mri 

- 1 and | - - + m (77 + 277 ), 
&) a) 

which are the constants in the exponential factor common to all functions of 
the same class, consider a function, of positive class m with a single accidental 
singularity, in the form 

* m (z, u) = eW 



<r(u- 6 X ) o- (u - &). <r (u - b m+1 ) <r(z-u) 
where b 1} b. 2 , ..., b m are arbitrary constants, of sum s, and 
m (&> + 2ft) ) = 6 OT+1 + fcj + b.> + . . . b m - u 

= b m+l +s-u. 
The function y- m satisfies the equations 

_mirzi 

y- w (z + 2<w, u) = i/r m (z, u), y, tt (z + 2&) , w) = e~ -^ m (z, u) ; 

regarded as a function of z, it has u for its sole accidental singularity, 
evidently simple. 

The function - can be expressed in the form 

I \I/* I It I 

u k) . . . a- (u b m ) o- {s m (&) 



(r^-b,) ............ a-(z-b m ) a{u- z-s + m(a> + 2~w 7 )} 

Regarded as a function of u, it has z, \, . . ., b m for zeros and z + s - m (to + 2o> ) 
for its sole accidental singularity, evidently simple : also 

z + &J + ...+ b m - {z + s - m (&) + 2o/)} = m (w + 2o> ). 

Hence owing to the values of X arid p, it follows that -- } - x when re- 

f> m (*, tt) 

garded as a function of u, satisfies all the conditions that establish a doubly- 
periodic function of the third kind of positive class m, so that 

1 1 



~i 7 ~ =r ^ 

and therefore 



mnz 

ty m (z, u + 2o>) = ^ m (z, u), ^ m (z, u + 20) ) = e~ijr m (z, u). 
Evidently -f m (z, u) regarded as a function of u is of negative class m : its 
infinities and its sole zero can at once be seen from the form 

-b m ) o-{u-z-s+m(ca 



<r(u -z)*^-^)...*^- b m ) a- {s - m (to + 2o) )j 

Each of the infinities is simple. In the vicinity of u = z, the expansion of 
the function is 

^^ z + positive integral powers of u z : 

19 



290 TERTIARY FUNCTIONS [145. 

and, in the vicinity of u = b r , it is 

C* ( 7\ 

r j + positive integral powers of u b r , 

Lv "~~ \Jrp 

where G r (z) denotes 

r) <r(z-bi)--.<r(z-br-i)<r(z-br +l )...a(z-b m )o-{z+s-br-m(a>+2a> )} 
a- (b r - 6j). ..cr (b r - 6 r _!) cr(b r - b r+l )...cr(b r - b in ) o-[s-ra(eo + 2&> )} 

and is therefore an integral function of z of positive class m. 

Let 4> (14) be a doubly-periodic function of the third kind, of positive class 
m ; and let its irreducible accidental singularities, that is, those which occur 
in a parallelogram containing the point u, be a^ of order !+/*!, a., of order 
1 + ju, 2 , and so on. In the immediate vicinity of a point a r , let 



--... 



\ 

- 



rr r;r -... r -,-~- - - r r. 

cm du- du^J u a,. 

Then proceeding as in the case of the secondary doubly-periodic functions 
( 137), we construct a function 

F(u) = 3?(u)^ m (z, u). 
We at once have F (u + 2o>) = F (u) = F(u + 2a> ), 

so that F(u) is a doubly-periodic function of the first kind; hence the sum 
of its residues for all the poles in a parallelogram of periods is zero. 

For the infinities of F (u), which arise through the factor ty m (z, u}, wea 
have as the residue for u = z 

-<*>(*), 
and as the residue for u = b r , where r = 1, 2, ..., m, 



In the vicinity of a,., we have 

fy n (Z, u) = ^r m (Z, r ) + (u - Or) ty m (z, O. r ) 



where dashes imply differentiation of ^r m {z, u} with regard to u, after which 
u is made equal to a,. ; so that in <I> (u) ty m (z, u) the residue for u = a r , where 
r = l, 2, ..., is 

E r (z) = A r ,jr m (z, ct r ) + B, Tjr m (z, a,.) + C r ty m " (z, a r ) + ...+ M r <^ m ^r) ( z> a r \ 
Hence we have 



and therefore (z)= 2 E,(z)+ 2 <& (b r ) G r (z), 

s=l r=l 

giving the expression of <l> (z) by means of doubly -periodic functions of tht 
third kind, which are of positive class m and have either no accidental singu-> 
larity or only one and that a simple singularity. 



145.] OF NEGATIVE CLASS 291 

The m quantities b lt ..., b m are arbitrary; the simplest case which occurs 
is when the m zeros of &(z) are different and are chosen as the values 
of &!,..., b m . The value of 3>(z) is then 

<&(*)= 2 JS .C*), 

s=l 

where the summation extends to all the irreducible accidental singularities ; 
while, if there be the further simplification that all the accidental singularities 
are simple, then 

<I> (z) = A 1 TJr m (2, !> + A s ty m (z, ot 2 ) + . . ., 

the summation extending to all the irreducible simple singularities. 
The quantity ty m (z, a r ), which is equal to 

) <r(z-bd...<r(z- b m ) <r{z + 2b-m(<o + 2ft/) - a r ] 



a-(a r b 1 )...a- (a r - b m ) <r {26 - m (co + 2ft> )} a- (z - a r ) 

and is subsidiary to the construction of the function E (z\ is called the 
simple element of positive class m. 

In the general case, the portion 



gives an integral function of z, and the portion 2 E s (z) gives a fractional 

s=l 

function of z. 

146. Secondly, let m be negative and equal to n. The equations 
satisfied by & (z} are 



i = <I> 0), <I> (z + 2ft) ) = e w < 0), 

and the number of irreducible singularities is greater by n than the number 
of irreducible zeros. 

One expression for <i> (z} is at once obtained by forming its reciprocal, 
which satisfies the equations 

11 1 -2-** i 

f\ /K / -\ > 



and is therefore of the class just considered: the value of is of the 

q>(^) 

form 

ZE s ( z ) + ^A r G r (z}. 

For purposes of expansion, however, this is not a convenient form as it gives 
only the reciprocal of <I> (z}. 

To represent the function, Appell constructed the element 



TT s v Ffr-K-*Wl 7r(2 
gr * . cot *- 



192 



292 TERTIARY FUNCTIONS [146. 

which, since the real part of to fan is positive, converges for all values of z and 

y, except those for which 

z = y (mod. 2&>, 2&> ). 

For each of these values one term of the series, and therefore the series 
itself, becomes infinite of the first order. 

Evidently % (z, y + 2o>) = % M (z, y}, 

niryi 

Xn (z, y + 2eo ) = e %(*, y); 
therefore in the present case 

0(y)=*(3f)jfr (**?)> 

regarded as a function of ^/, is a doubly-periodic function of the first kind. 

Hence the sum of the residues of its irreducible accidental singularities 
is zero. 

When the parallelogram is chosen, which includes z, these singularities 
are 

(i) y = z, arising through % n (z, y} ; 

(ii) the singularities of < (y}, which are at least n in number, and are 
n + I when <& has I irreducible zeros. 

The expansion of Xn 0> y), in powers of y - z, in the vicinity of the point 
z, is 

+ positive integral powers of y z ; 



y-z 
therefore the residue of II (y) is 

Let ct r be any irreducible singularity, and in the vicinity of a,, let 3> (y) denote 

d 



-I- positive integral powers of y Or, 

where the series of negative powers is finite because the singularity is 
accidental ; then the residue of H (y} is 

A r ^ (Z, Or) + B r X n (*, r) + C r % ,/ (z, Ct,) + . . . + P r X* 0> > )> 

where % n (A) (^, a r ) is the value of 

d x % n (z, y) 

dy* 

when y = 0r after differentiation. Similarly for the residues of other singu 
larities : and so, as their sum is zero, we have 

< (Z) = 2 {A r X n (*, r) + B r X n (*, ,) + ...+ P, X n W (?, r)}, 

the summation extending over all the singularities. 



146.] OF NEGATIVE CLASS 293 

The simplest case occurs when all the N(>n) singularities a are accidental 
and of the first order ; the function 4> (z) can then be expressed in the form 

A l Xn (Z, i) + A 2 Xn (Z, Oj) + . . . + A N Xn (z, #) 

The quantity Xn (z, a), which is equal to 

T *^" ^p{(-i) +} TT - a 

a 2/6 COt -^. 



. 

2(0 



is called the simple element for the expression of a doubly-periodic function of 
the third kind of negative class n. 



Ex. Deduce the result 



_ ^ ( iVcot 
TT snu s= -oo v I 2K / 

147. The function Xn (z, y} can be used also as follows. Since Xm (z, y), 
qua function of y, satisfies the equations 

% m (z, 11 + 2(i)} = Y (z, 7/\ 
llv \ s {/ / /V //fc \ J ts / 

miryi 

Xm (z, y + 2o/) = e~^x m (z, y), 

which are the same equations as are satisfied by a function of y of positive 
class m, therefore Xm (<*> z), which is equal to 



2 e cot 



being a function of z, satisfies the characteristic equations of 142 ; and, in 
the vicinity of z = a, 

Xm ( a > z ) - + positive integral powers of z a. 

Z ~~" OC 

If then we take the function 4> (z) of 145, in the case when it has simple 
singularities at a lt 2 , ... and is of positive class m, then 



4> (z) + A, x w (a, , 

is a function of positive class m without any singularities: it is therefore 
equal to an integral function of positive class m, say to G(z) t where 

G (z) = Ae^+^a- (z-a l }...(r(z- a m ), 
so that 3>(z) = G(z)-A 1 Xm(ct 1 ,2)-A,x m (<Xt,z)-.... 

Ex. As a single example, consider a function of negative class 2, and let it have no 
zero within the parallelogram of reference. Then for the function, in the canonical 
product-form of 143, the two irreducible infinities are subject to the relation 



and the function is * (z) = AV "V" - 

o- (z Cj) o- (z-c 2 ) 



294 TERTIARY FUNCTIONS [147. 

The simple elements to express 3> (z) as a sum are 

2.<!iri , , 

{{s-lX + Cl} ,77, 

* " rt (s-C! -2*,), 



4iri, ,, 
7T -(ci-<o) - r-w-c i TT 

= _ e <-> 2 e a> cot (2 + j-2no) 



after an easy reduction, 

4irj 



The residue of *(s) for c n which is a simple singularity, is 

Us-( 
A l = Ktf a v< 

and for c 2 , also a simple singularity, it is 



, 

so that ^- = -e w =-e w 

^2 

Hence the expression for 4> (z) as a sum, which is 



! 

becomes A l ( X2 (2, Cj) - e u ^2 (^ - c i)} 

that is, it is a constant multiple of 



Again, 



j - - 

<r(z- GJ) a- (z + c^ - 2o>- 



on changing the constant factor. Hence it is possible to determine L so that 



ni Tti 

" C c - e<a 



Taking the residues of the two sides for z=c 1} we have 
and therefore finally we have 



-C]*- Ci -- C, 

Le < = e > 



>-.>-* 



TtlC 



(a (s, c) - e w X 2 ( 2 > - c ) 



<* C ot^ L (2-c 1 -2su) )-e w cot - - (z + c x - 2s w ) K 

2<a 2a> ) 

the right-hand side of which admits of further modification if desired. 



147.] PSEUDO-PERIODIC FUNCTIONS 295 

Many examples of such developments in trigonometrical series are given by Hermite*, 
Biehlerf, HalphenJ, Appell, and Krause||. 

148. We shall not further develop the theory of these uniform doubly- 
periodic functions of the third kind. It will be found in the memoirs of 
Appell to whom it is largely due; and in the treatises of Halphen**, and 
of Rausenberger"f"f. 

It need hardly be remarked that the classes of uniform functions of a 
single variable which have been discussed form only a small proportion of 
functions reproducing themselves save as to a factor when the variable 
is subjected to homographic substitutions, of which a very special example 
is furnished by linear additive periodicity. Thus there are the various 
classes of pseudo-automorphic functions, ( 305) called Thetafuchsian by Pom- 
care, their characteristic equation being 



for all the substitutions of the group determining the function : and other 
classes are investigated in the treatises which have just been quoted. 

The following examples relate to particular classes of pseudo-periodic 
functions. 

Ex. 1. Shew that, if F (z) be a uniform function satisfying the equations 



m 

where b is a primitive mth root of unity, then F(z) can be expressed in the form 



where f(z) denotes the function 



and prove that \F(z)dz can be expressed in the form of a doubly-periodic function 
together with a sum of logarithms of doubly-periodic functions with constant coefficients. 

(Goursat.) 

* Comptes Rendus, t. Iv, (1862), pp. 1118. 

t Sur les developpements en series des fonctions doublement periodiqucs de troisieme espece, 
(These, Paris, Gauthier-Villars, 1879). 

Traite des fonctions elliptiques, t. i, chap. xm. 

Annales de VEc. Norm. Sup., 3 rae S6r., t. i, pp. 135164, t. ii, pp. 936, t. iii, pp. 942. 

|| Math. Ann., t. xxx, (1887), pp. 425436, 516534. 

* Traite des fonctions elliptiques, t. i, chap. xiv. 

ft Lehrbuch der Theorie der periodischen Functional, (Leipzig, Teubner, 1884), where further 
references are given. 



296 PSEUDO-PERIODIC FUNCTIONS [148. 

Ex. 2. Shew that, if a pseudo-periodic function be denned by the equations 



and if, in the parallelogram of periods containing the point z, it have infinities c, ... such 
that in their immediate vicinity 



then/ (2) can be expressed in the form 

- ^ ^{^I+ ...... +,}>, 

the summation extending over all the infinities of/ (z) in the above parallelogram of periods, 
and the constants (7 15 ... being subject to the condition 

+ iVS C l = A o> X o. 

Deduce an expression for a doubly-periodic function <f) (z) of the third kind, by 
assuming 

/W-f]8. (Halphen.) 

(f> \g) 

Ex. 3. If S(z) be a given doubly-periodic function of the first kind, then a 
pseudo-periodic function F(z), which satisfies the equations 

F(z + ^} = F(z), 
mriz 
F (z + 2o> ) = e ~"~ S (z} F (z), 

where n is an integer, can be expressed in the form 



where -4 is a constant and TT (2) denotes 



the summation extending over all points &,. and the constants B r being subject to the 
relation 



Explain how the constants b, G and B can be determined. (Picard.) 

Ex. 4. Shew that the function F(z) defined by the equation 

for values of \z\, which are <1, satisfies the equation 

and that the function F l (a!)=^ ^rjr-i 

where (j)(,v) = 3? 1, and </>(.* )> f ()r positive and negative values of n, denotes (/> [0 {< (#)}] 

<f> being repeated n times, and a is the positive root of a 3 a - 1 = ; satisfies the equation 

for real values of the variable. 

Discuss the convergence of the series which defines the function F l (x). (Appell.) 



CHAPTER XIII. 

FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM. 

149. WE may consider at this stage an interesting set* of important 
theorems, due to Weierstrass, which are a justification, if any be necessary, 
for the attention ordinarily (and naturally) paid to functions belonging to 
the three simplest classes of algebraic, simply-periodic and doubly-periodic 
functions. 

A function <f> (u) is said to possess an algebraical addition theorem, when 
among the three values of the function for arguments u, v, and u + v, where u 
and v are general and not merely special arguments, an algebraical equation 
exists f having its coefficients independent of u and v. 

150. It is easy to see, from one or two examples, that the function does 
not need to be a uniform function of the argument. The possibility of 
multiformity is established in the following proposition : 

A function defined by an algebraical equation, the coefficients of which are 
uniform algebraical functions of the argument, or are uniform simply -periodic 
functions of the argument, or are uniform doubly -periodic functions of the 
argument, possesses an algebraical addition-theorem. 

* They are placed in the forefront of Schwarz s account of Weierstrass s theory of elliptic 
functions, as contained in the Formeln und Lehrsdtze zum Gebrauche der elliptischen Functionen; 
but they are there stated ( 13) without proof. The only proof that has appeared is in a 
memoir by Phragmen, Acta Math., t. vii, (1885), pp. 3342; and there are some statements 
(pp. 390393) in Biermann s Theorie der analytischen Functionen relative to the theorems. The 
proof adopted in the text does not coincide with that given by Phragme n. 

t There are functions which possess a kind of algebraical addition -theorem ; thus, for 
instance, the Jacobian Theta-functions are such that e A (u + w) O^ (u- v) can be rationally ex 
pressed in terms of the Theta-functions having it and v for their arguments. Such functions 
are, however, naturally excluded from the class of functions indicated in the definition. 

Such functions, however, possess what may be called a multiplication-theorem for multipli 
cation of the argument by an integer, that is, the set of functions 6 (nut) can be expressed 
algebraically in terms of the set of functions 6 (M). This is an extremely special case of a set 
of transcendental functions having a multiplication-theorem, which are investigated by Poincare, 
Liouville, 4" S6r., t. iv, (1890), pp. 313365. 



298 EXAMPLES OF FUNCTIONS [150. 

First, let the coefficients be algebraical functions of the argument u. If 
the function defined by the equation be U, we have 

U m g (u) + U m ~ l gi (u) + ...+g m (u) = 0, 

where g (u),gi(u}, ...,g m (u) are rational integral algebraical functions of u 
of degree, say, not higher than n. The equation can be transformed into 

u n f/U\+ u 1 - 1 /! ( U) + ... + f n ( U) = 0, 

where f (U), fi(U), > fn(U) are rational integral algebraical functions of 
U of degree not higher than m. 

If V denote the function when the argument is v, and W denote it when 
the argument is u + v, then 

w/ (7) + ^ 1 / 1 (7) + ... +f n (V) M 0, 
and (u + v) n / ( W) + (u + vY^f, ( W ) + . . . +f n ( W ) = 0. 

The algebraical elimination of the two quantities u and v between these 
three equations leads to an algebraical equation between the quantities 
/(/"), /(7) and f (W), that is, to an algebraical equation between U, V, W, 
say of the form 

G(U, V, F) = 0, 

where G denotes an algebraical function, with coefficients independent of 
u and v. It is easy to prove that G is symmetrical in U and 7, and that 
its degree in each of the three quantities U, 7, W is wn 2 . The equation 
G = implies that the function U possesses an algebraical addition- theorem. 

Secondly, let the coefficients* be uniform simply-periodic functions of 
the argument u. Let &> denote the period: then, by 113, each of these 

TT IL 

functions is a rational algebraical function of tan . Let u denote 

tan ; then the equation is of the form 

U m g (u ) + U m ^g, (u } + ...+ g m 00 = 0, 

where the coefficients g are rational algebraical (and can be taken as 
integral) functions of u . If p be the highest degree of u in any of them, 
then the equation can be transformed into 

u vfo ( U) + u P- 1 /! ( U) + . . . + f p ( U) = 0, 

where f (U), fi(U), ..., f p (U) are rational integral algebraical functions of 
U of degree not higher than m. 

* The limitation to uniformity for the coefficients has been introduced merely to make the 
illustration simpler; if in any case they were multiform, the equation would be replaced by 
another which is equivalent to all possible forms of the first arising through the (finite) 
multiformity of the coefficients : and the new equation would conform to the specified 
conditions. 



150.] POSSESSING AN ADDITION-THEOREM 299 

Let v denote tan , and w denote tan -- ; then the corresponding 
cy &) 

values of the function are determined by the equations 



and w *>f (W) + w p-*/! (W) + ... +f p (W) = 0. 

The relation between u , v , w is 

u v w + u + v - w = 0. 

The elimination of the three quantities u , v , w among the four equations 
leads as before to an algebraical equation 

G(U, V, W) = 0, 

where G denotes an algebraical function (now of degree mp 2 ) with coefficients 
independent of u and v. The function U therefore possesses an algebraical 
addition-theorem. 

Thirdly, let the coefficients be uniform doubly-periodic functions of the 
argument u. Let &> and &/ be the two periods ; and let @ (u), the Weier- 
strassian elliptic function in those periods, be denoted by . Then every 
coefficient can be expressed in the form 



~L 

where L, M, N are rational integral algebraical functions of f of finite 
degree. Unless each of the quantities N is zero, the form of the equation 
when these values are substituted for the coefficients is 

A+Bp (u) = 0, 

so that A* = &(?-g-9*)\ 

and this is of the form 

U m ff* () + U ^g, (|) + . . . + g m () - 0, 

where the coefficients g are rational algebraical (and can be taken as integral) 
functions of If q be the highest degree of in any of them, the equation 
can be transformed into 



where the coefficients / are rational integral algebraical functions of U of 
degree not higher than 2m. 

Let TJ denote $ (v) and f denote p(u + v); then the corresponding values 
of the function are determined by the equations 

......... +f q (V)=0, 



By using Ex. 4, 131, it is easy to shew that the relation between , rj, is 



300 WEIERSTRASS S THEOREM ON FUNCTIONS [150. 

The elimination of , ij, from the three equations leads as before to an 

algebraical equation 

G(U,V, W) = 0, 

of finite degree and with coefficients independent of u and v. Therefore in this 
case also the function U possesses an algebraical addition-theorem. 

If, however, all the quantities N be zero, the equation defining U is of the 

form 

U m h () + U^h, () + . . . + h m () = 0, 

and a similar argument then leads to the inference that U possesses an 
algebraical addition-theorem. 

The proposition is thus completely established. 

151. The generalised converse of the preceding proposition now suggests 
itself : what are the classes of functions of one variable that possess an alge 
braical addition-theorem? The solution is contained in Weierstrass s theorem : 

An analytical function <f> (u), which possesses an algebraical theorem, is 
either 

(i) an algebraical function of u ; or 

liru 

(ii) an algebraical function of e , where w is a suitably chosen 
constant ; or 

(iii) an algebraical function of the elliptic function %>(u), the periods or 
the invariants g. z and g 3 being suitably chosen constants. 

Let U denote </> (w). 

For a given general value of u, the function U may have m values where, 
for functions in general, there is not a necessary limit to the value of m ; it 
will be proved that, when the function possesses an algebraical addition- 
theorem, the integer m must be finite. 

For a given general value of U, that is, a value of U when its argument is 
not in the immediate vicinity of a branch-point if there be branch-points, the 
variable u may have p values, where p may be finite or may be infinite. 

Similarly for given general values of v and of V, which will be used to 
denote < (v). 

First, let p be finite. Then because u has p values for a given value of U 
and v has p values for a given value of V, and since neither set is affected by the 
value of the other function, the sum u + v has p 2 values because any member of 
the set u can be combined with any member of the set v ; and this number 
p 2 of values of u + v is derived for a given value of U and a given value of V. 

Now in forming the function <j>(u + v), which will be denoted by W, we 
have m values of W for each value of u + v and therefore we have mp 2 values 
of W for the whole set, that is, for a given value of U and a given value of V. 



151.] POSSESSING AN ADDITION-THEOREM 301 

Hence the equation between U, V, W is of degree* mp 2 in W, necessarily 
finite when the equation is algebraical ; and therefore m is finite. 

Because m is finite, U has a finite number m of values for a given value of 
u ; and, because p is finite, u has a finite number p of values for a given value of 
U. Hence U is determined in terms of u by an algebraical equation of degree 
m, the coefficients of which, are rational integral algebraical functions of 
degree p ; and therefore U is an algebraic function of u. 

152. Next, let p be infinite ; then (see Note, p. 303) the system of values 
may be composed of (i) a single simply-infinite series of values or (ii) a finite 
number of simply-infinite series of values or (iii) a simply-infinite number of 
simply-infinite series of values, say, a single doubly-infinite series of values or 
(iv) a finite number of doubly-infinite series of values or (v) an infinite 
number of doubly-infinite series of values where, in (v), the infinite number 
is not restricted to be simply-infinite. 

Taking these alternatives in order, we first consider the case where the p 
values of u for a given general value of U constitute a single simply -infinite 
series. They may be denoted by f (u, n), where n has a simply-infinite 
series of values and the form of/ is such that f(u, 0) = u. 

Similarly, the p values of v for a given general value of V may be denoted 
by/(y, n), where n has a simply-infinite series of values. Then the different 
values of the argument for the function W are the set of values given by 

f(u,n)+f(v,ri), 

for the simply-infinite series of values for n and the similar series of values 
for n . 

The values thus obtained as arguments of W must all be contained in 
the series f(u + v, n"}, where n" has a simply-infinite series of values ; and, 
in the present case,/(w + w, n"} cannot contain other values. Hence for some 
values of n and some values of n , the total aggregate being not finite, the 
equation 

f(u,n}+f(v,n }=f(u + v,n") 
must hold, for continuously varying values of u and v. 

In the first place, an interchange of u and v is equivalent to an interchange 
of n and n on the left-hand side; hence n" is symmetrical in n and n . 
Again, we have 

df(u, n) _ df(u + v, n") 
du 3 (u + v) 



dv 

* The degree for special functions may be reduced, as in Cor. 1, Prop. XIII, 118; but in no 
case is it increased. Similarly modifications, in the way of finite reductions, may occur in the 
succeeding cases ; but they will not be noticed, as they do not give rise to essential modification 
in the reasoning. 



302 FORM OF ARGUMENT [152. 

so that the form of f(u, n) is such that its first derivative with regard to u is 
independent of u. Let (n) be this value, where (n), independent of u, may 
be dependent on n ; then, since 



we have f(u, n) = uO (n) + ty (n), 

-fy- (n) being independent of u. Substituting this expression in the former 

equation, we have the equation 

u6 (n) + ^ (n) + v9 (n } + f (71 ) = (u + v)6 (n"} + ^ (n"), 
which must be true for all values of u and v ; hence 

e(n)=e(n") = d(n ), 

so that 6 (n) is a constant and equal to its value when n = 0. But when n is 
zero,/(w, 0) is u ; so that 9 (0) = 1 and ^ (0) = 0, and therefore 

f(u, n) = u + Tjr (n), 
where i/r vanishes with n. 

The equation defining ty is 



for values of n from a singly-infinite series and for values of n from the same 
series, that series is reproduced for TO". Since ^ (n) vanishes with n, we take 

^ (n) = HX (n), 

and therefore rc% (n) + n % (n ) = ri x (n"). 

Again, when n vanishes, the required series of values of n" is given by taking 
n" = n ; and, when n does not vanish, n" is symmetrical in n and n , so that 

we have 

n" = n + n + nn\, 

where X is not infinite for zero or finite values of n or n . Thus 
HX (n) + n x (n) = (n + TO + -nw X) % (w + ?* + wi X). 

Since the left-hand side is the sum of two functions of distinct and inde 
pendent magnitudes, the form of the equation shews that it can be satisfied 

only if 

X = 0, so that n" = n + n ; 

and % 0) = % ( n// ) 

= %(n \ 
so that each is a constant, say o> ; then 

f(u, n} = u + nco, 

which is the form that the series must adopt when the series f(u + v, n") is 
obtained by the addition of/(, n) and/0, n )- 



152.] IN A SIMPLY-INFINITE SERIES 303 

It follows at once that the single series of arguments for W is obtained, 
as one simply-infinite series, of the form u + v+n"a). For each of these 
arguments we have m values of W, and the set of m values of W is 
the same for all the different arguments; that is, W has m values for a 
given value of U and a given value of V. Moreover, U has m values for each 
argument and likewise V; hence, as the equation between U, V, W is of 
a degree that is necessarily finite because the equation is algebraical, the 
integer m is finite. 

It thus appears that the function U has a finite number m of values for 
each value of the argument u, and that for a given value of the function the 
values of the argument form a simply-periodic series represented by u + nw. 

But the function tan ( ) is such that, for a given value, the values of the 

V 03 J 

argument are represented by the series u + nw ; hence for each value of 

tan ( 1 there are m values of U and for each value of U there is one value 
\ o / 

of tan -- . It therefore follows, by SS 113, 114, that between U and tan ( } 
w \ to / 

there is an algebraical relation which is of the first degree in tan - - and the 



O) 

U 



rath degree in U, that is, U is an algebraic function of tan - . Hence U is 



(I) 



an algebraic function also of e <" . 

Note. This result is based upon the supposition that the series of argu 
ments, for which a branch of the function has the same value, can be arranged 
in the form/(w, n), where n has a simply-infinite series of integral values. If, 
however, there were no possible law of this kind the foregoing proof shews 
that, if there be one such law, there is only one such law, with a properly 
determined constant co then the values would be represented by u l} u, ...,u p 
with p infinite in the limit. In that case, there would be an infinite number of 
sets of values for u + v of the type W A + v^, where X and p might be the same 
or might be different ; each set would give a branch of the function W and then 
there would be an infinite number of values of W corresponding to one branch 
of U and one branch of V. The equation between U, V and W would be of 
infinite degree in W, that is, it would be transcendental and not algebraical. 
The case is excluded by the hypothesis that the addition-theorem is alge 
braical, and therefore the equation between U, V and W is algebraical. 

153. Next, let there be a number of simply-infinite series of values of 
the argument of the function, say q, where q is greater than unity and 
may be either finite or infinite. Let u l} u. 2 , ..., u q denote typical members 
of each series. 

Then all the members of the series containing u l must be of the form 



304 FORM OF ARGUMENT [153. 

fi ( u i> n )> f r an infinite series of values of the integer n. Otherwise, as in the 
preceding note, the sum of the values in the series of arguments u and of 
those in the same series of arguments v would lead to an infinite number of 
distinct series of values of the argument u + v, with a corresponding infinite 
number of values W ; and the relation between U, V, W would cease to be 
algebraical. 

In the same way, the members of the corresponding series containing ^ 
must be of the form/! (v 1} ri) for an infinite series of values of the integer n . 
Among the combinations 



the simply-infinite series fi(tii+v 1} n") must occur for an infinite series 
of values of n"; and therefore, as in the preceding case, 

fi(u ly n) = M 1 + nw 1 , 

where toj is an appropriate constant. Further, there is only one series of 
values for the combination of these two series ; it is represented by 

Ui + v 1 + n"w l . 

In the same way, the members of the series containing u 2 can be repre 
sented in the form u 2 + nco 2 , where o> 2 is an appropriate constant, which may 
be (but is not necessarily) the same as Wj ; and the series containing u. 2 , 
when combined with the set containing v 2 , leads to only a single series 
represented in the form u. 2 + v 2 + ri o) 2 . And so on, for all the series in order. 

But now since u 2 + m 2 a) 2 , where m 2 is an integer, is a value of u for a given 
value of U, it follows that U (u 2 + ra 2 a> 2 ) = U (w 2 ) identically, each being equal 

to U. Hence 

U (M! + m l w l + 7n.,<y 2 ) = U (i^ + ra^) = U (u^ = U, 

and therefore ^ + m l (a l + ra 2 &> 2 is also a value of u for the given value of U, 
leading to a series of arguments which must be included among the original 
series or be distributed through them. Similarly u 1 + 2m r (i) r , where the 
coefficients ra are integers and the constants to are properly determined, 
represents a series of values of the variable u, included among the original 
series or distributed through them. And generally, when account is taken of 
all the distinct series thus obtained, the aggregate of values of the variable u 
can be represented in the form Wx+2w r tu r , for \ 1, 2, ..., K, where K is 
some finite or infinite integer. 

Three cases arise, (a) when the quantities are equal to one another or 
can be expressed as integral multiples of only one quantity a>, (6) when the 
quantities &> are equivalent to two quantities f^ and O 2 (the ratio of which is 
not real), so that each quantity &> can be expressed in the form 

a> r =p lr fi l +p ar si a> 

the coefficients p lr , p 2r being finite integers ; (c) when the quantities are 
not equivalent to only two quantities, such as flj and fl 2 . 



153.] SIMPLY-PERIODIC FUNCTIONS 305 

For case (a), each of the K infinite series of values u can be expressed 
in the form u^+pci), for X = 1, 2, ..., and integral values of p. 

First, let K be finite, so that the original integer q is finite. Then the 
values of the argument for W are of the type 



that is, M A + ?V +>"&>, 

for all combinations of \ and fju and for integral values of p". There are thus 
K- series of values, each series containing a simply-infinite number of terms 
of this type. 

For each of the arguments in any one of these infinite series, W has ra 
values ; and the set of m values is the same for all the arguments in one and 
the same infinite series. Hence W has w/c 2 values for all the arguments in 
all the series taken together, that is, for a given value of U and a given 
value of V. The relation between U, V, W is therefore of degree m 2 , 
necessarily finite when the equation is algebraical ; hence m is finite. 

It thus appears that the function U has a finite number m of values for 
each value of the argument u, and that for a given value of the function there 
are a finite number K of distinct series of values of the argument of the form 

7TU 

u+poi), w being the same for all the series. But the function tan -- has 
one value for each value of u and the series u+pat represents the series of 

7TU 

values of u for a given value of tan . It therefore follows that there are 

CO 

m values of U for each value of tan and that there are K values of tan 

to o> 

for each value of U ; and therefore there is an algebraical relation between 

U and tan , which is of degree K in the latter and of degree m in the 
&) 

iiru 
TTlI 

former. Hence U is an algebraic function of tan and therefore also of e M . 



Next, let K be infinite, so that the original integer q is infinite. Then, 
as in the Note in 152, the equation between U, V, W will cease to be 
algebraical unless each aggregate of values u^+pw, for each particular 
value of p and for the infinite sequence X= 1, 2, ..., K, can be arranged in a 
system or a set of systems, say a in number, each of the form f p (u+pa), p p ) 
for an infinite series of values of p p . Each of these implies a series of values 
fp(v+p u>, p p ) of the argument of V for the same series of values of p p as of 
p p> and also a series of values f p (u + v+p"(o, p p ") of the argument of W for 
the same series of values of p p ". By proceeding as in 152, it follows that 

f p (u +pa>, pp} = u+pto +p p (0p, 

where &> p is an appropriate constant, the ratio of which to &> can be proved 
F. 20 



306 FORM OF ARGUMENT [153. 

(as in 106) to be not purely real, and p p has a simply-infinite succession of 
values. The integer a may be finite or it may be infinite. 

When ay and all the constants o> which thus arise are linearly equivalent 
to two quantities f^ and O 2 , so that the terms additive to u can be expressed 
in the form 8^ + s. 2 fl, then the aggregate of values u can be expressed 
in the form 



for a simply-infinite series for p l and for p 2 ; and p has a series of values 
1, 2, ..., <r. This case is, in effect, the same as case (6). 

When o) and all the constants are not linearly equivalent to only 
two quantities, such as Oj and IL>, we have a case which, in effect, is the 
same as case (c). 

These two cases must therefore now be considered. 

For case (6), either as originally obtained or as derived through parfc 
of case (a), each of the (doubly) infinite series of values of u can be expressed 
in the form 



for X = 1, 2, ..., <r and for integral values of _p, and p,. The integer a may be 
finite or infinite ; the original integer q is infinite. 

First, let cr be finite. Then the values of the argument for W are of the 
type 



that is, u\ + v^ +pi"li + p 2 "O 2 , 

for all combinations of \ and p and for integral values of >/ and p.". There 
are thus cr 2 series of values, each series containing a doubly-infinite number ofl 
terms of this type. 

For every argument there are m values of W ; and the set of m values is 
the same for all the arguments in one and the same infinite series. Thus W 
has mo- 2 values for all the arguments in all the series, that is, for a given value 
of U and a given value of V; and it follows, as before, from the consideration i 
of the algebraical relation, that m is finite. 

The function U thus has m values for each value of the argument u ; and 
for a given value of the function there are cr series of values of the argument, 
each series being of the form w x + PI^I +p. 2 Q*- 

Take a doubly-periodic function having Oj and H 2 for its periods, such* 1 
that for a given value of the values of its arguments are of the foregoing 
form. Whatever be the expression of the function, it is of the order cr. , 
Then U has m values for each value of @, and @ has one value for each . 
value of U; hence there is an algebraical equation between U and , ow 

* All that is necessary for this purpose is to construct, by the use of Prop. XII, 118, ai 
function having, as its irreducible simple infinities, a series of points aj, a 2 ,..., a<7 special* 
values of j, w 2 , ..., u a in the parallelogram of periods, chosen so that no two of the <r points a 
coincide. 



153.] DOUBLY-PERIODIC FUNCTIONS 307 

:he first degree in the latter and of the rath degree in U: that is, U is an 
algebraical function of @. But, by Prop. XV. 119, can be expressed in 
the form 



where L, M, N are rational integral algebraical functions of $ (u), if f^ and H 2 
be the periods of g) (u); and g) (u) is a two- valued algebraical function of jjp (u), 
so that is an algebraical function of i@ (u). Hence also U is an algebraical 
function of $(u\ the periods o/<p (u) being properly chosen. 

This inference requires that a, the order of , be greater than 1. 
Because U has m values for an argument u, the symmetric function St/" 
has one value for an argument u and it is therefore a uniform function. 
But each term of the sum has the same value for u+pifli+pfl t as for 
u ; and therefore this uniform function is doubly-periodic. The number of 
independent doubly-infinite series of values of u for a uniform doubly- 
periodic function is at least two : and therefore there must be at least two 
doubly-infinite series of values of u, so that <r > 1. Hence a function, that 
possesses an addition-theorem, cannot have only one doubly-infinite series of 
values for its argument. 

If cr be infinite, there is an infinite series of values of u of the form 
+ p^ + p.fl z ; an argument, similar to that in case (a), shews that this is, 
in effect, the same as case (c). 

It is obvious that cases (ii), (iii) and (iv) of 152 are now completely 
covered ; case (v) of 152 is covered by case (c) now to be discussed in 154. 

154. For case (c), we have the series of values u represented by a number 
of series of the form 



where the quantities &> are not linearly equivalent to two quantities flj and 
Q 2 - The original integer q is infinite. 

Then, by 108, 110, it follows that integers m can be chosen in an 
unlimited variety of ways so that the modulus of 



r=l 

is infinitesimal, and therefore in the immediate vicinity of any point u^ 
there is an infinitude of points at which the function resumes its value. 
Such a function would, as in previous instances, degenerate into a mere 
constant ; and therefore the combination of values which gives rise to this 
case does not occur. 

All the possible cases have been considered: and the truth of Weierstrass s 

202 



308 EXAMPLES [154. 

theorem* that a function, which has an algebraical addition-theorem, is either 

imi 

an algebraical function of u, or of e " (where &> is suitably chosen), or of g> (u), 
where the periods of @(u) are suitably chosen, is established; and it has 
incidentally been established it is, indeed, essential to the derivation of the 
theorem that a function, which has an algebraical addition-theorem, has only 
a finite number of values for a given argument. 

It is easy to see that the first derivative has only a finite number of values 
for a given argument; for the elimination of U between the algebraical 
equations 



, , 

leads to an equation in U of the same finite degree as G in U. 

Further, it is now easy to see that if the analytical function < (u), which 
possesses an algebraical addition-theorem, be uniform, then it is a rational 

iiru 

function either of u, or of e w , or of $> (u) and $ (u) ; and that any uniform 
function, which is transcendental in the sense of 47 and which possesses an 
algebraical addition-theorem, is either a simply-periodic function or a doubly- 
periodic function. 

The following examples will illustrate some of the inferences in regard to the number 
of values of <p (u + v) arising from series of values for u and v. 

Ex. I. Let U=u* + (2u+l)*. 

Evidently m, the number of values of U for a value of u, is 4 ; and, as the rationalised 
form of the equation is 



the value of p, being the number of values of u for a given value of U, is 2. Thus the 
equation in W should be, by 151, of degree (4.2 2 ) 16. 

This equation is n {3 ( W 2 - U 2 - F 2 ) + 1 - 2k r } = 0, 

HI 
where k r is any one of the eight values of 

W(2W*-I)*+U(2U*-l$+V(2V*-l)*; 



an equation, when rationalised, of the 16th degree in W. 

Ex. 2. Let U=cosu. 

Evidently m = l; the values of u for a given value of U are contained in the double 
series u + 2irn, -u + 2irn, for all values of n from -QO to +GO. The values of u + v are 
, that is, u + v + 27rp; -u + 27rn+v + 2irm, that is, -u + v + 2-n-p ; 
, that is, u-v + ^Trp; -u + 2irn-v + 2irm, that is, -u-v + Znp, 



* The theorem has been used by Schwarz, Ges. Werke, t. ii, pp. 260268, in determining all 
the families of plane isothermic cirrves which are algebraical curves, an isothermic curve being 
of the form u = c, where w is a function satisfying the potential-equation 



154.] THE DIFFERENTIAL EQUATION 309 

to that the number of series of values of u+v is four, each series being simply-infinite. 
It might thus be expected that the equation between U, V, W would be of degree 

4 = ) 4 in W ; but it happens that 

cos (u + v)=cos( -u-v), 
and so the degree of the equation in W is reduced to half its degree. The equation is 

W 2 - 2 WU V+ U 2 + V 2 - 1 = 0. 

Ex. 3. Let U=&iiu. 

Evidently m = l; and there are two doubly-infinite series of values of u determined 
by a given value of U, having the form u + 2ma> + 2m <o , o> - w + 2mo> + 2m V. Hence the 
values of u + v are 

= u+v (mod. 2c0, 2o> ) ; = ca-u + v (mod. 2, 2 ) ; 
= ca + u-v(mod. 2o>, 2<o ) ; = -u-v (mod. 2o>, 2&> ) ; 

four in number. The equation may therefore be expected to be of the fourth degree 
in W; it is 

4 (1 - 6 T2 ) (1 - F 2 ) (1 - IF 2 ) = (2 - U 2 - F 2 - IF 2 +2*7272 W 2^ 

155. But it must not be supposed that any algebraical equation between 
U, V, W, which is symmetrical in U and V, is one necessarily implying the 
representation of an algebraical addition-theorem. Without entering into a 
detailed investigation of the formal characteristics of the equations that are 
suitable, a latent test is given by implication in the following theorem, also 
due to Weierstrass : 

If an analytical function possess an algebraical addition-theorem, an 
algebraical equation involving the function and its first derivative with regard 
to its argument exists ; and the coefficients in this equation do not involve the 
argument of the function. 

The proposition might easily be derived by assuming the preceding 
proposition, and applying the known results relating to the algebraical 
dependence between those functions, the types of which are suited to the 
representation of the functions in question, and their derivatives ; we shall, 
however, proceed more directly from the equation expressing the algebraical 
addition-theorem in the form 

G(U,V, F) = 0, 

which may be regarded as a rationally irreducible equation. 
Differentiating with regard to u, we have 

WU +MW^Q 

dU L + dW 
and similarly, with regard to v, we have 

a> + *<=<>, 

from which it follows that 



310 EXPRESSION OF [155. 

This equation* will, in general, involve W; in order to obtain an equation 
free from W, we eliminate W between 

n A a ^^ rr/ d6r Tr/ 

G = and ^j- U = V , 

oil ov 

the elimination being possible because both equations are of finite degree; 
and thus in any case we have an algebraical equation independent of W and 
involving U, U , V, V. 

Not more than one equation can arise by assigning various values to v, a 
quantity that is independent of u ; for we should have either inconsistent 
equations or simultaneous equations which, being consistent, determine a! 
limited number of values of U and U for all values of u, that is, only a 
number of constants. Hence there can be only one equation, obtained by 
assigning varying values to v; and this single equation is the algebraical 
equation between the function and its first derivative, the coefficients being 
independent of the argument of the function. 

Note. A test of suitability of an algebraical equation G between 
three variables U, V, W to represent an addition-theorem is given by the 
condition that the elimination of W between 

G-Q and U ^-V 

dU~ dV 

leads to only a single equation between U and U for different values of V 
and V. 

Ex. Consider the equation 

(Z-U- V- W)*-4(1-U}(1- F)(l- F) = 0. 
The deduced equation involving U 1 and V is 

(2FTF- V- W+ U} U = (2UW- U- W+ V) V, 

, th-it W (V-U}(V +U } 

= (SV~lTU r 

The elimination of W is simple. We have 



_ 

(27-1) U -(2U-\) F" 

F U -l-U V 



utd 2 U V W- 

( 

Neglecting 4 (F+ U 1) = 0, which is an irrelevant equation, arid multiplying by 
(2F 1) U (2Ul) F , which is not zero unless the numerator also vanish, and this 
would make both U and V zero, we have 

( F+ U- 1) {(1 - F) U - (1 - U} F } 2 = (1 - U) (1 - F) ( U - F ) (2 F- 1) U - (2 U- 1) F }, 
and therefore V(U-V}(1- V] (7 2 + U( F- U} (1 - U} F 2 = 0. 

It is permissible to adopt any subsidiary irrational or non-algebraical form as the equivalent 
of G = 0, provided no special limitation to the subsidiary form be implicitly adopted. Thus, if W 
can be expressed explicitly in terms of U and F, this resoluble (but irrational) equivalent of the 
equation often leads rapidly to the equation between U and its derivative. 



155.] THE ADDITION-THEOREM 311 

When the irrelevant factor U- V is neglected, this equation gives 

U * F 2 

U(l-U}~ V(l - V) 

the equation required : and this, indeed, is the necessary form in which the equation 
involving U and U arises in general, the variables being combined in associate pairs. 
Each side is evidently a constant, say 4a 2 ; and then we have 



Then the value of U is sin 2 (aM+/3), the arbitrary additive constant of integration 
being /3 ; by substitution in the original equation, (3 is easily proved to be zero. 

156. Again, if the elimination between 

a - o and U - V 

a du u ~w v 

be supposed to be performed by the ordinary algebraical process for finding 

o/~y o/^r 

the greatest common measure of G and U %Tf V %-\r> regarded as functions 

of W, the final remainder is the eliminant which, equated to zero, is the 
differential equation involving U, U , V, F ; and the greatest common measure, 
equated to zero, gives the simplest equation in virtue of which the equations 

G = and ^y U = _-^ V subsist. It will be of the form 
oil ov 

f(W,U,V, U ,V ) = 0. 

If the function have only one value for each value of the argument, so that it 
is a uniform function, this last equation can give only one value for W , for all 
the other magnitudes that occur in the equation are uniform functions of 
their respective arguments. Since it is linear in W, the equation can be 
expressed in the form 

W = R(U, V, U , V \ 

where R denotes a rational function. Hence* : 

A uniform analytical function (f> (u), which possesses an algebraical 
addition-theorem, is such that (f> (u + v) can be expressed rationally in terms 
of $ (u), < (w), $ (v) and <j> (v). 

It need hardly be pointed out that this result is not inconsistent with the 
fact that the algebraical equation between ( (u + v), (f> (u) and <f> (v) does not, 
in general, express $(u + v) as a rational function of (f> (u) and <f>(v). And it 
should be noticed that the rationality of the expression of < (u + v) in terms 
of <j) (u), $ (v), (/> (w), $ (v) is characteristic of functions with an algebraical 
addition-theorem. Instances do occur of functions such that <j)(u + v) can be 
expressed, not rationally, in terms of < (u), </> (v), </> (u), </> (v) ; they do not 
possess an algebraical addition-theorem. Such an instance is furnished by 
%(u) , the expression of (u + v), given in Ex. 3 of 131, can be modified so 
as to have the form indicated. 

* The theorem is due to Weierstrass ; see Schwarz, 2, (I.e. in note to p. 297). 



CHAPTER XIV. 

CONNECTION OF SURFACES. 

157. IN proceeding to the discussion of multiform functions, it was 
stated ( 100) that there are two methods of special importance, one of which 
is the development of Cauchy s general theory of functions of complex vari 
ables and the other of which is due to Riemann. The former has been 
explained in the immediately preceding chapters ; we now pass to the 
consideration of Riemann s method. But, before actually entering upon it, 
there are some preliminary propositions on the connection of surfaces which 
must be established ; as they do not find a place in treatises on geometry, an 
outline will be given here but only to that elementary extent which is 
necessary for our present purpose. 

In the integration of meromorphic functions, it proved to be convenient 
to exclude the poles from the range of variation of the variable by means of 
infinitesimal closed simple curves, each of which was thereby constituted a 
limit of the region : the full boundary of the region was composed of the 
aggregate of these non-intersecting curves. 

Similarly, in dealing with some special cases of multiform functions, it 
proved convenient to exclude the branch-points by means of infinitesimal 
curves or by loops. And, in the case of the fundamental lemma of 16, the 
region over which integration extended was considered as one which possibly 
had several distinct curves as its complete boundary. 

These are special examples of a general class of regions, at all points 
within the area of which the functions considered are monogeiiic, finite, and 
continuous and, as the case may be, uniform or multiform. But, important 
as are the classes of functions which have been considered, it is necessary to 
consider wider classes of multiform functions and to obtain the regions which 
are appropriate for the representation of the variation of the variable in each 
case. The most conspicuous examples of such new functions are the algebraic 
functions, adverted to in 94 99 ; and it is chiefly in view of their value 
and of the value of functions dependent upon them, as well as of the kind of 
surface on which their variable can be simply represented, that we now 
proceed to establish some of the topological properties of surfaces in general. 

158. A surface is said to be connected when, from any point of it to any 
other point of it, a continuous line can be drawn without passing out of the 



158.] 



EXAMPLES OF CONNECTED SURFACES 



313 



surface. Thus the surface of a circle, that of a plane ring such as arises in 
Lambert s Theorem, that of a sphere, that of an anchor-ring, are connected 
surfaces. Two non-intersecting spheres, not joined or bound together in any 
manner, are not a connected surface but are two different connected surfaces. 
It is often necessary to consider surfaces, which are constituted by an 
aggregate of several sheets ; but, in order that the surface may be regarded 
as connected, there must be junctions between the sheets. 

One of the simplest connected surfaces is such a plane area as is enclosed 
and completely bounded by the circumference of a circle. All lines drawn in 
it from one internal point to another can be deformed into one another ; any 
simple closed line lying entirely within it can be deformed so as to be 
evanescent, without in either case passing over the circumference ; and any 
simple line from one point of the circumference to another, when regarded as 
an impassable barrier, divides the surface into two portions. Such a surface 
is called* simply connected. 

The kind of connected surface next in point of simplicity is such a plane 
area as is enclosed between and is completely bounded by the circumferences 
of two concentric circles. All lines in the surface 
from one point to another cannot necessarily be 
deformed into one another, e.g., the lines z az and 
zj)z; a simple closed line cannot necessarily be 
deformed so as to be evanescent without crossing 
the boundary, e.g., the line az^bza ; and a simple 
line from a point in one part of the boundary to 
a point in another and different part of the 
boundary, such as a line AB, does not divide the 
surface into two portions but, set as an impassable barrier, it makes the 
surface simply connected. 

Again, on the surface of an anchor-ring, a closed line can be drawn in 
two essentially distinct ways, abc, cib c , such 
that neither can be deformed so as to be evanes 
cent or so as to pass continuously into the other. 
If abc be made the only impassable barrier, a 
line such as afty cannot be deformed so as to be 
evanescent ; if ab c be made the only impassable 
barrier, the same holds of a line such as a/3 y . 
In order to make the surface simply connected, 
two impassable barriers, such as abc and ab c , 
must be set. 

Surfaces, like the flat ring or the anchor- 




Fig. 35. 




Fig. 36. 



* Sometimes the term vionadelphic is used. The German equivalent is einfach ziisammen- 
hangend. 



314 



CROSS-CUTS AND LOOP-CUTS 



[158. 



ring, are called* multiply connected] the establishment of barriers has made it 
possible, in each case, to modify the surface into one which is simply connected. 

159. It proves to be convenient to arrange surfaces in classes according 
to the character of their connection ; and these few illustrations suggest that 
the classification may be made to depend, either upon the resolution of the 
surface, by the establishment of barriers, into one that is simply connected, 
or upon the number of what may be called independent irreducible circuits. 
The former mode that of dependence upon the establishment of barriers 
will be adopted, thus following Biemann-f- ; but whichever of the two modes 
be adopted (and they are not necessarily the only modes) subsequent de 
mands require that the two be brought into relation with one another. 

The most effective way of securing the impassability of a barrier is to 
suppose the surface actually cut along the line of the barrier. Such a section 
of a surface is either a cross-cut or a loop-cut. 

If the section be made through the interior of the surface from one point 





Fig. 37. 

of the boundary to another point of the boundary, without intersecting itself 
or meeting the boundary save at its extremities, it is called a cross-cut\. 
Every part of it, as it is made, is to be regarded as boundary during the 
formation of the remainder ; and any cross-cut, once made, is to be regarded 
as boundary during the formation of any cross-cut subsequently made. 
Illustrations are given in Fig. 37. 

The definition and explanation imply that the surface has a boundary. 
Some surfaces, such as a complete sphere and a complete anchor-ring, do not 
possess a boundary; but, as will be seen later ( 163, 168) from the 
discussion of the evanescence of circuits, it is desirable to assign some 
boundary in order to avoid merely artificial difficulties as to the numerical 

* Sometimes the term polyadc.lphic is used. The German equivalent is mehrfach zusammen- 
Mngcnd. 

t " Grundlagen fur eine allgemeine Theorie der Functionen einer veriindeiiichen complexen 
Grosse," Eiemann s Gesammelte Werke, pp. 9 12; "Theorie der Abel schen Functionen," ib.,/ 
pp. 8489. When reference to either of these memoirs is made, it will be by a citation "et ih^ 
page or pages in the volume of lliemann s Collected Works. 

This is the equivalent used for the German word Querschnitt ; French writers use Section, 
and Italian writers use Trasversale or Taglio trasversale. 



159.] CONNECTION DEFINED 315 

expression of the connection. This assignment usually is made by taking for 
the boundary of a surface, which otherwise has no boundary, an infinitesimal 
closed curve, practically a point; thus in the figure of the anchor-ring 
(Fig. 36) the point a is taken as a boundary, and each of the two cross-cuts 
begins and ends in a. 

If the section be made through the interior of the surface from a point 
not on the boundary and, without meeting the boundary or crossing itself, 
return to the initial point, (so that it has the form of a simple curve lying 





Fig. 38. 

entirely in the surface), it is called* a loop-cut. Thus a piece can be cut 
out of a bounded spherical surface by a loop-cut (Fig. 38) ; but it does 
not necessarily give a separate piece when made in the surface of an 
anchor-ring. 

It is evident that both a cross-cut and a loop-cut furnish a double 
boundary-edge to the whole aggregate of surface, whether consisting of two 
pieces or of only one piece after the section. 

Moreover, these sections represent the impassable barriers of the pre 
liminary explanations ; and no specified form was assigned to those barriers. 
It is thus possible, within certain limits, to deform a cross-cut or a loop-cut 
continuously into a closely contiguous and equivalent position. If, for 
instance, two barriers initially coincide over any finite length, one or other 
can be slightly deformed so that finally they intersect only in a point ; the 
same modification can therefore be made in the sections. 

The definitions of simple connection and of multiple connection will nowf* 
be as follows : 

A surface is simply connected, if it be resolved into two distinct pieces by 
every cross-cut; but if there be any cross-cut, which does not resolve it into 
distinct pieces, the surface is multiply connected. 

160. Some fundamental propositions, relating to the connection of 
surfaces, may now be derived. 

* This is the equivalent used for the German word Riickkehrsclmitt ; French writers use the 
word Retroscction. 

t Other definitions will be required, if the classification of surfaces be made to depend on 
methods other than resolution by sections. 



316 RESOLUTION BY CROSS-CUTS [160. 

I. Each of the two distinct pieces, into which a simply connected surface S 
is resolved by a cross-cut, is itself simply connected. 

If either of the pieces, made by a cross-cut ab, be not simply connected, 
then some cross-cut cd must be possible which will not resolve that piece into 
distinct portions. 

If neither c nor d lie on ab, then the obliteration of the cut ab will restore 
the original surface 8, which now is not resolved by the cut cd into distinct 
pieces. 

If one of the extremities of cd, say c, lie on ab, then the obliteration of the 
portion cb will change the two pieces into a single piece which is the original 
surface 8; and 8 now has a cross-cut acd, which does not resolve it into 
distinct pieces. 

If both the extremities lie on ab, then the obliteration of that part of ab 
which lies between c and d will change the two pieces into one ; this is the 
original surface 8, now with a cross-cut acdb, which does not resolve it into 
distinct pieces. 

These are all the possible cases should either of the distinct pieces of 8 
not be simply connected ; each of them leads to a contradiction of the simple 
connection of 8 , therefore the hypothesis on which each is based is untenable, 
that is, the distinct pieces of 8 in all the cases are simply connected. 

COROLLARY 1. A singly connected surface is resolved by n cross-cuts into 
Ti+1 distinct pieces, each simply connected; and an aggregate of m simply 
connected surfaces is resolved by n cross-cuts into n -f m distinct pieces each 
simply connected. 

COROLLARY 2. A surface that is resolved into two distinct simply con 
nected pieces by a cross-cut is simply connected before the resolution. 

COROLLARY 3. // a multiply connected surface be resolved into two 
different pieces by a cross-cut, both of these pieces cannot be simply connected. 

We now come to a theorem* of great importance : 

II. If a resolution of a surface by m cross-cuts into n distinct simply 
connected pieces be possible, and also a different resolution of the same surface by 
fjb cross-cuts into v distinct simply connected pieces, then m n = fj, v. 

Let the aggregate of the n pieces be denoted by 8 and the aggregate of 
the v pieces by 2 : and consider the effect on the original surface of a united 
system of in + p simultaneous cross-cuts made up of the two systems of the 
m and of the /j, cross-cuts respectively. The operation of this system can be 
carried out in two ways : (i) by effecting the system of /u, cross-cuts on 8 and 

* The following proof of this proposition is substantially due to Neumann, p. 157. Another 
proof is given by Riemann, pp. 10, 11, and is amplified by Durege, Elemente der Theorie der 
Functional, pp. 183 190 ; and another by Lippich, see Durege, pp. 190 197. 






160.] CONNECTIVITY 317 

(ii) by effecting the system of m cross-cuts on 2 : with the same result on the 
original surface. 

After the explanation of 159, we may justifiably assume that the lines 
of the two systems of cross-cuts meet only in points, if at all : let 8 be the 
number of points of intersection of these lines. Whenever the direction of a 
cross-cut meets a boundary line, the cross-cut terminates ; and if the direction 
continue beyond that boundary line, that produced part must be regarded as 
a new cross-cut. 

Hence the new system of /u, cross-cuts applied to S is effectively equiva 
lent to (j, + & new cross-cuts. Before these cuts were made, S was composed 
of n simply connected pieces ; hence, after they are applied, the new arrange 
ment of the original surface is made up of n + (/j, + 8) simply connected 
pieces. 

Similarly, the new system of m cross-cuts applied to 2 will give an 
arrangement of the original surface made up of v + (m + 8) simply connected 
pieces. These two arrangements are the same : and therefore 

n + fj, + 8 v + in + 8, 
so that m n = p v. 

It thus appears that, if by any system of q cross-cuts a multiply connected 
surface be resolved into a number p of pieces distinct from one another and 
all simply connected, the integer q p is independent of the particular 
system of the cross-cuts and of their configuration. The integer qp is 
therefore essentially associated with the character of the multiple connection 
of the surface : and its invariance for a given surface enables us to arrange 
surfaces according to the value of the integer. 

No classification among the multiply connected surfaces has yet been 
made : they have merely been defined as surfaces in which cross-cuts can be 
made that do not resolve the surface into distinct pieces. 

It is natural to arrange them in classes according to the number of cross 
cuts which are necessary to resolve the surface into one of simple connection 
or a number of pieces each of simple connection. 

For a simply connected surface, no such cross-cut is necessary: then 
q = 0, p=l, and in general q p = l. We shall say that the connectivity* 
is unity. Examples are furnished by the area of a plane circle, and by a 
spherical surface with one hole^. 

A surface is called doubly- connected when, by one appropriate cross-cut, 
the surface is changed into a single surface of simple connection : then q = 1, 
p = 1 for this particular resolution, and therefore in general, qp = Q. We 

* Sometimes order of connection, sometimes adelphic order ; the German word, that is used, 
is Grundzahl. 

+ The hole is made to give the surface a boundary ( 163). 



318 EFFECT OF CROSS-CUTS [160. 

shall say that the connectivity is 2. Examples are furnished by a plane ring 
and by a spherical surface with two holes. 

A surface is called triply-connected when, by two appropriate cross-cuts, 
the surface is changed into a single surface of simple connection : then q = 2, 
p = l for this particular resolution and therefore, in general, q p = l. We 
shall say that the connectivity is 3. Examples are furnished by the surface 
of an anchor- ring with one hole in it*, and by the surfaces -f- in Figure 39, the 
surface in (2) not being in one plane but one part beneath another. 




Fig. 39. 

And, in general, a surface will be said to be ^V-ply connected or its 
connectivity will be denoted by N, if, by N 1 appropriate cross-cuts, it can 
be changed into a single surface that is simply connected |. For this 
particular resolution q = N\, p = l: and therefore in general 

q-p = N-2, 
or N = q-p + 2. 

Let a cross-cut I be drawn in a surface of connectivity N. There are 
two cases to be considered, according as it does not or does divide the surface 
into distinct pieces. 

First, let the surface be only one piece after I is drawn : and let its 
connectivity then be N . If in the original surface q cross-cuts (one of 
which can, after the preceding proposition, be taken to be I) be drawn 
dividing the surface into p simply connected pieces, then 

N = q-p+ 2. 

To obtain these p simply connected pieces from the surface after the cross-cut 
I, it is evidently sufficient to make the q 1 original cross-cuts other than I ; 
that is, the modified surface is such that by q 1 cross-cuts it is resolved into 
p simply connected pieces, and therefore 



Hence N = N 1, or the connectivity of the surface is diminished by unity. 

* The hole is made to give the surface a boundary ( 163). 

t Riemann, p. 89. 

J A few writers estimate the connectivity of such a surface as N- 1, the same as the number 
of cross-cuts which can change it into a single surface of the simplest rank of connectivity : the 
estimate in the text seems preferable. 



160.] 



ON THE CONNECTIVITY 



319 



Secondly, let the surface be two pieces after I is drawn, of connectivities 
Ni and N 2 respectively. Let the appropriate JVj 1 cross-cuts in the former, 
and the appropriate N 2 1 in the latter, be drawn so as to make each a 
simply connected piece. Then, together, there are two simply connected 
pieces. 

To obtain these two pieces from the original surface, it will suffice to 
make in it the cross-cut I, the Ni I cross-cuts, and the N 2 l cross-cuts, 
that is, 1 + (Ni. 1) + (N* 1) or Nj, + N 2 1 cross-cuts in all. Since these, 
when made in the surface of connectivity N, give two pieces, we have 



and therefore 

If one of the pieces be simply connected, the connectivity of the other is JV; 
so that, if a simply connected piece of surface be cut off a multiply connected 
surface, the connectivity of the remainder is unchanged. Hence : 

III. If a cross-cut be made in a surface of connectivity N and if it do 
not divide it into separate pieces, the connectivity of the modified surface is 
Nl; but if it divide the surface into two separate pieces of connectivities N! 
and N, then N l + N 2 = N+ 1. 




Illustrations are shewn, in Fig. 40, of the effect of cross-cuts on the two 
surfaces in Fig. 39. 

IV. In the same way it may be proved that, if s cross-cuts be made in a 
surface of connectivity N and divide it into r+l separate pieces (where r^.s) 
of connectivities N 1} N 2 , ..., N r+l respectively, then 



a more general result including both of the foregoing cases. 

Thus far we have been considering only cross-cuts : it is now necessary 
to consider loop-cuts, so far as they affect the connectivity of a surface in 
which they are made. 




320 EFFECT OF LOOP-CUTS [160. 

A loop-cut is changed into a cross-cut, if from A any point of it a cross-cut 
be made to any point C in a boundary-curve of the 
original surface, for CAbdA (Fig. 41) is then evi- / 

dently a cross-cut of the original surface ; and CA is 
a cross-cut of the surface, which is the modification 
of the original surface after the loop-cut has been 
made. Since, by definition, a loop-cut does not 
meet the boundary, the cross-cut CA does not 
divide the modified surface into distinct pieces ; 
hence, according as the effect of the loop-cut is, \ Fi 8- 41 - 

or is not, that of making distinct pieces, so will 
the effect of the whole cross-cut be, or not be, that of making distinct pieces. 

161. Let a loop-cut be drawn in a surface of connectivity N; as before 
for a cross-cut, there are two cases for consideration, according as the loop-cut 
does or does not divide the surface into distinct pieces. 

First, let it divide the surface into two distinct pieces, say of connectivities 
N! and N 2 respectively. Change the loop-cut into a cross-cut of the original 
surface by drawing a cross-cut in either of the pieces, say the second, from a 
point in the course of the loop-cut to some point of the original boundary. 
This cross-cut, as a section of that piece, does not divide it into distinct 
pieces: and therefore the connectivity is now N? (= N 2 1). The effect of 
the whole section, which is a single cross-cut, of the original surface is to 
divide it into two pieces, the connectivities of which are JV a and N 2 : hence, 
by S 160, III., 



and therefore N 1 + N a 

If the piece cut out be simply connected, say JVj. = 1, then the connectivity 
of the remainder is N + 1. But such a removal of a simply connected piece 
by a loop-cut is the same as making a hole in a continuous part of the 
surface : and therefore the effect of making a simple hole in a continuous part 
of a surface is to increase by unity the connectivity of the surface. 

If the piece cut out be doubly connected, say N : = 2, then the connect 
ivity of the remainder is N, the same as the connectivity of the original 
surface. Such a portion would be obtained by cutting out a piece with a 
hole in it which, so far as concerns the original surface, would be the same as 
merely enlarging the hole an operation that naturally would not affect 
the connectivity. 

Secondly, let the loop -cut not divide the surface into two distinct pieces : 
and let N be the connectivity of the modified surface. In this modified 
surface make a cross-cut k from any point of the loop-cut to a point of the 
boundary: this does not divide it into distinct pieces and therefore the 
connectivity after this last modification is N -I. But the surface thus 



161.] ON THE CONNECTIVITY 321 

finally modified is derived from the original surface by the single cross-cut, 
constituted by the combination of k with the loop-cut : this single cross-cut 
does not divide the surface into distinct pieces and therefore the connectivity 
after the modification is N 1. Hence 



that is, JV = N, or the connectivity of a surface is not affected by a loop-cut 
which does not divide the surface into distinct pieces. 

Both of these results are included in the following theorem : 

V. If after any number of loop-cuts made in a surface of connectivity 
N, there be r + 1 distinct pieces of surface, of connectivities JV^ JV 2 , ..., N r+lt 
then 

N, + N 3 + ...... + JV r+1 = JV+2r. 

Let the number of loop-cuts be s. Each of them can be changed into a 
cross-cut of the original surface, by drawing in some one of the pieces, as may 
be convenient, a cross-cut from a point of the loop-cut to a point of a 
boundary ; this new cross-cut does not divide the piece in which it is drawn 
into distinct pieces. If k such cross-cuts (where k may be zero) be drawn in 
the piece of connectivity N m , the connectivity becomes N m , where 

N N~ If- 

" m * m I" j 
r+l r+l r+l 

hence 2 N m = 2 N m -2k= X N m - s. 

m=\ m-\ m=l 

We now have s cross-cuts dividing the surface of connectivity JV into r + l 
distinct pieces, of connectivities JV/, JV/, ..., JV/, N r+1 ; and therefore, by 
160, IV., 



so that JVj + JV 2 + . . . 4- N r+1 = JV + 2r. 

This result could have been obtained also by combination and repetition 
of the two results obtained for a single loop-cut. 

Thus a spherical surface with one hole in it is simply connected : when 
n l other different holes* are made in it, the edges of the holes being 
outside one another, the connectivity of the surface is increased by n 1, 
that is, it becomes n. Hence a spherical surface with n holes in it is n-ply 
connected. 

162. Occasionally, it is necessary to consider the effect of a slit made in 
the surface. 

If the slit have neither of its extremities on a boundary (and therefore no 
point on a boundary) it can be regarded as the limiting form of a loop-cut 
which makes a hole in the surface. Such a slit therefore ( 161) increases the 
connectivity by unity. 

* These are holes in the surface, not holes bored through the volume of the sphere ; one of 
the latter would give two holes in the surface. 

F- 21 



BOUNDARIES [162. 

If the slit have one extremity (but no other point) on a boundary, it can 
be regarded as the limiting form of a cross-cut, which returns 
on itself as in the figure, and cuts off a single simply con- / 
nected piece. Such a slit therefore ( 160, III.) leaves the 
connectivity unaltered. 

If the slit have both extremities on boundaries, it ceases \ 
to be merely a slit : it is a cross-cut the effect of which on Fl 8- 42 - 

the connectivity has been obtained. We do not regard such 
sections as slits. 

163. In the preceding investigations relative to cross-cuts and loop-cuts, 
reference has continually been made to the boundary of the surface con 
sidered. 

The boundary of a surface consists of a line returning to itself, or of a 
system of lines each returning to itself. Each part of such a boundary-line 
as it is drawn is considered a part of the boundary, and thus a boundary-line 
cannot cut itself and pass beyond its earlier position, for a boundary cannot 
be crossed: each boundary-line must therefore be a simple curve*. 

Most surfaces have boundaries : an exception arises in the case of closed 
surfaces whatever be their connectivity. It was stated ( 159) that a 
boundary is assigned to such a surface by drawing an infinitesimal simple 
curve in it or, what is the same thing, by making a small hole. The 
advantage of this can be seen from the simple example of a spherical 
surface. 

When a small hole is made in any surface the connectivity is increased 
by unity : the connectivity of the spherical surface after the hole is made is 
unity, and therefore the connectivity of the complete spherical surface 
must be taken to be zero. 

The mere fact that the connectivity is less than unity, being that of the 
simplest connected surfaces with which we have to deal, 
is not in itself of importance. But let us return for a 
moment to the suggested method of determining the 
connectivity by means of the evanescence of circuits 
without crossing the boundary. When the surface is 
the complete spherical surface (Fig. 43), there are two 
essentially distinct ways of making a circuit C evan 
escent, first, by making it collapse into the point a, Fig. 43. 
secondly by making it expand over the equator and 

then collapse into the point b. One of the two is superfluous : it introduces 
an element of doubt as to the mode of evanescence unless that mode be 
specified a specification which in itself is tantamount to an assignment of 

* Also a line not returning to itself may be a boundary ; it can be regarded as the limit of a 
simple curve when the area becomes infinitesimal. 





163.] EFFECT OF CROSS-CUTS ON BOUNDARIES 323 

boundary. And in the case of multiply connected surfaces the absence of 
boundary, as above, leads to an artificial reduction of the connectivity by 
unity, arising not from the greater simplicity of the surface but from the 
possibility of carrying out in two ways the operation of reducing any circuit 
to given circuits, which is most effective when only one way is permissible. 
We shall therefore assume a boundary assigned to such closed surfaces as in 
the first instance are destitute of boundary. 

164. The relations between the number of boundaries and the connect 
ivity of a surface are given by the following propositions. 

I. The boimdary of a simply connected surface consists of a single line. 

When a boundary consists of separate lines, then a cross-cut can be made 
from a point of one to a point of another. By proceeding from 
P, a point on one side of the cross-cut, along the boundary 
ac...cVwe can by a line lying wholly in the surface reach a 
point Q on the other side of the cross-cut : hence the parts of 
the surface on opposite sides of the cross-cut are connected. 
The surface is therefore not resolved into distinct pieces by the 
cross-cut. 

A simply connected surface is resolved into distinct pieces Fig. 44. 
by each cross-cut made in it : such a cross-cut as the foregoing 
is therefore not possible, that is, there are not separate lines which make up 
its boundary. It has a boundary : the boundary therefore consists of a single 
line. 

II. A cross-cut either increases by unity or diminishes by unity the number 
of distinct boundary -lines of a multiply connected surface. 

A cross-cut is made in one of three ways : either from a point a of one 
boundary-line A to a, point b of another boundary-line B ; or from a point a 
of a boundary-line to another point a of the same boundary-line ; or from a 
point of a boundary-line to a point in the cut itself. 

If made in the first way, a combination of one edge of the cut, the 
remainder of the original boundary A, the other edge of the cut and the 
remainder of the original boundary B taken in succession, form a single 
piece of boundary ; this replaces the two boundary-lines A and B which 
existed distinct from one another before the cross-cut was made. Hence the 
number of lines is diminished by unity. An example is furnished by a plane 
ring (ii., Fig. 37, p. 314). 

If made in the second way, the combination of one edge of the cut with 
the piece of the boundary on one side of it makes one boundary-line, and the 
combination of the other edge of the cut with the other piece of the boundary 
makes another boundary-line. Two boundary-lines, after the cut is made, 

212 



324 NUMBER OF BOUNDARY-LINES [164. 

replace a single boundary-line, which existed before it was made : hence the 
number of lines is increased by unity. Examples are furnished by the cut 
surfaces in Fig. 40, p. 319. 

If made in the third way, the cross-cut may be considered as constituted 
by a loop-cut and a cut joining the loop-cut to the boundary. The boundary- 
lines may now be considered as constituted (Fig. 41, p. 320) by the closed 
curve ABD and the closed boundary abda c e ...eca; that is, there are now 
two boundary-lines instead of the single boundary-line ce...e c c in the uncut 
surface. Hence the number of distinct boundary-lines is increased by unity. 

COROLLARY. A loop-cut increases the number of distinct boundary-lines 
by two. 

This result follows at once from the last discussion. 

III. The number of distinct boundary-lines of a surface of connectivity N 
is N 2k, where k is a positive integer that may be zero. 

Let m be the number of distinct boundary-lines ; and let N 1 appro 
priate cross-cuts be drawn, changing the surface into a simply connected 
surface. Each of these cross-cuts increases by unity or diminishes by unity 
the number of boundary-lines ; let these units of increase or of decrease be 
denoted by e^ e 2 , ..., #_!. Each of the quantities e is + 1 ; let k of them be 
positive, and N 1 k negative. The total number of boundary-lines is 

therefore 

m + k-(N-l-k). 

The surface now is a single simply connected surface, and there is therefore 
only one boundary-line ; hence 

m + k-(N-l-k) = l, 
so that m = N 2k ; 

and evidently k is an integer that may be zero. 

COROLLARY 1. A closed surface with a single boundary-line* is of odd 
connectivity. 

For example, the surface of an anchor-ring, when bounded, is of con 
nectivity 3; the surface, obtained by boring two holes through the volume 
of a solid sphere, is, when bounded, of connectivity 5. 

If the connectivity of a closed surface with a single boundary be 2p + 1, 
the surface is often said-f- to be of class p ( 178, p. 349.) 

COROLLARY 2. If the number of distinct boundary lines of a surface of 
connectivity N be N, any loop-cut divides the surface into two distinct pieces. 

After the loop-cut is made, the number of distinct boundary-lines is 
N+2; the connectivity of the whole of the cut surface is therefore not less 

* See 159. 

t The German word is Geschlecht ; French writers use the word genre, and Italians genere. 



164.] LHUILIER S THEOREM 325 

than N+2. It has been proved that a loop-cut, which does not divide the 
surface into distinct pieces, does not affect the connectivity ; hence as the 
connectivity has been increased, the loop-cut must divide the surface into 
two distinct pieces. It is easy, by the result of 161, to see that, after the 
loop-cut is made, the sum of connectivities of the two pieces is N+2, so 
that the connectivity of the whole of the cut surface is equal to N + 2. 

Note. Throughout these propositions, a tacit assumption has been made, 
which is important for this particular proposition when the surface is the 
means of representing the variable. The assumption is that the surface is 
bifacial and not unifacial ; it has existed implicitly throughout all the 
geometrical representations of variability : it found explicit expression in 
4 when the plane was brought into relation with the sphere : and a cut 
in a surface has been counted a single cut, occurring in one face, though it 
would have to be counted as two cuts, one on each side, were the surface 
unifacial. 

The propositions are not necessarily valid, when applied to unifacial 
surfaces. Consider a surface made out of a long rectangular slip of paper, 
which is twisted once (or any odd number of times) and then has its ends 
fastened together. This surface is of double connectivity, because one 
section can be made across it which does not divide it into separate pieces ; 
it has only a single boundary-line, so that Prop. III. just proved does not 
! apply. The surface is unifacial ; and it is possible, without meeting the 
boundary, to pass continuously in the surface from a point P to another 
point Q which could be reached merely by passing through the material 
at P. 

We therefore do not retain unifacial surfaces for consideration. 

165. The following proposition, substantially due to Lhuilier*, may be 
taken in illustration of the general theory. 

If a closed surface of connectivity 2N + 1 (or of class N) be divided by 
circuits into any number of simply connected portions, each in the form of a 
curvilinear polygon, and if F be the number of polygons, E be the number of 
edges and S the number of angular points, then 

2N=2 + JE-F-S. 

Let the edges E be arranged in systems, a system being such that any 
lino in it can be reached by passage along some other line or lines of the 
system ; let k be the number of such systems -f. To resolve the surface into a 
number of simply connected pieces composed of the F polygons, the cross-cuts 
will be made along the edges ; and therefore, unless a boundary be assigned 

* Gergonne, Ann. de Math., t. iii, (1813), pp. 181186; see also Mobius, Ges. Werke, t. ii, 
p. 468. A circuit is defined in 166. 

t The value of k is 1 for the proposition and is greater than 1 for the Corollary. 



326 LHUILIER S THEOREM [165. 

to the surface in each system of lines, the first cut for any system will be a 
loop-cut. We therefore take k points, one in each system as a boundary ; 
the first will be taken as the natural boundary of the surface, and the 
remaining k\, being the limiting forms of k 1 infinitesimal loop-cuts, 
increase the connectivity of the surface by k 1, that is, the connectivity now 
is 2N+k. 

The result of the cross-cuts is to leave F simply connected pieces : hence 
Q, the number of cross-cuts, is given by 



At every angular point on the uncut surface, three or more polygons are 
contiguous. Let S m be the number of angular points, where m polygons are 
contiguous; then 



Again, the number. of edges meeting at each of the S 3 points is three, atl 
each of the $ 4 points is four, at each of the $ 5 points is five, and so on ; hence, 
in taking the sum 3$ 3 + 4$ 4 + 5$ 5 + . . ., each edge has been counted twice, once 
for each extremity. Therefore 



Consider the composition of the extremities of the cross-cuts ; the number 
of the extremities is 2Q, twice the number of cross-cuts. 

Each of the k points furnishes two extremities; for each such point 
is a boundary on which the initial cross-cut for each of the systems must 
begin and must end. These points therefore furnish 2k extremities. 

The remaining extremities occur in connection with the angular points. 
In making a cut, the direction passes from a boundary along an edge, past 
the point along another edge and so on, until a boundary is reached ; so that 
on the first occasion when a cross-cut passes through a point, it is made along 
two of the edges meeting at the point. Every other cross-cut passing through 
that point must begin or end there, so that each of the S 3 points will furnish 
one extremity (corresponding to the remaining one cross-cut through the 
point), each of the $ 4 points will furnish two extremities (corresponding to 
the remaining two cross-cuts through the point), and so on. The total 
number of extremities thus provided is 

S 3 + 2St+3S 5 + ... 
Hence 2Q = 2k + 8 3 + 2S t + 3S 6 + ... 



or Q = k + E-S, 

which combined with Q = 2N + k + F - 2, 
leads to the relation 2N=2 + E-F-S. 



165.] CIRCUITS ON CONNECTED SURFACES 327 

The simplest case is that of a sphere, when Euler s relation F + S = E + 2 
is obtained. The case next in simplicity is that of an anchor-ring, for which 
the relation is F+ S = E. 

COROLLARY. If the result of making the cross-cuts along the various edges 
be to give the F polygons, not simply connected areas but areas of connectivities 
jYj + 1, jV 2 + l, ..., Np+1 respectively, then the connectivity of the original 
surface is given by 



166. The method of determining the connectivity of a surface by means 
of a system of cross-cuts, which resolve it into one or more simply connected 
pieces, will now be brought into relation with the other method, suggested 
in 159, of determining the connectivity by means of irreducible circuits. 

A closed line drawn on the surface is called a circuit. 

A circuit, which can be reduced to a point by continuous deformation 
without crossing the boundary, is called reducible ; a circuit, which cannot be 
so reduced, is called irreducible. 

An irreducible circuit is either (i) simple, when it cannot without crossing 
the boundary be deformed continuously into repetitions of one or more 
circuits ; or (ii) multiple, when it can without crossing the boundary be 
deformed continuously into repetitions of a single circuit ; or (iii) compound, 
when it can without crossing the boundary be deformed continuously into 
combinations of different circuits, that may be simple or multiple. The 
distinction between simple circuits and compound circuits, that involve no 
multiple circuits in their combination, depends upon conventions adopted for 
each particular case. 

A circuit is said to be reconcileable with the system of circuits into a 
combination of which it can be continuously deformed. 

If a system of circuits be reconcileable with a reducible circuit, the 
system is said to be reducible. 

As there are two directions, one positive and the other negative, in which 
a circuit can be described, and as there are possibilities of repetitions and of 
compositions of circuits, it is clear that circuits can be represented by linear 
algebraical expressions involving real quantities and having merely numerical 
coefficients. 

Thus a reducible circuit can be denoted by 0. 

If a simple irreducible circuit, positively described, be denoted by a, the 
same circuit, negatively described, can be denoted by a. 

The multiple circuit, which is composed of m positive repetitions of the 
simple irreducible circuit a, would be denoted by ma ; but if the m repetitions 
were negative, the multiple circuit would be denoted by ma. 



328 CIRCUITS [106. 

A compound circuit, reconcileable with a system of simple irreducible 
circuits a 1} a 2 , ..., a n would be denoted by m 1 a 1 + m 2 a 2 -\- ... + m n a n , where 
mj, m 2 , ..., m n are positive or negative integers, being the net number of 
positive or negative descriptions of the respective simple irreducible circuits. 

The condition of the reducibility of a system of circuits a l , 2 , ..., a n , 
each one of which is simple and irreducible, is that integers m 1} m. 2 , ..., m n 
should exist such that 

m^j + m 2 a 2 + . . . + m n a n = 0, 

the sign of equality in this equation, as in other equations, implying that 
continuous deformation without crossing the boundary can change into one 
another the circuits, denoted by the symbols on either side of the sign. 

The representation of any compound circuit in terms of a system of 
independent irreducible circuits is unique : if there were two different 
expressions, they could be equated in the foregoing sense and this would 
imply the existence of a relation 

P& + p. 2 a 2 + . . . +p n a n = 0, 
which is excluded by the fact that the system is irreducible. 

Further, equations can be combined linearly, provided that the coefficients 
of the combinations be merely numerical. 

167. In order, then, to be in a position to estimate circuits on a multiply 
connected surface, it is necessary that an irreducible system of irreducible 
simple circuits should be known, such a system being considered complete 
when every other circuit on the surface is reconcileable with the system. 

Such a system is not necessarily unique ; and it must be proved that, if 
more than one complete system be obtainable, any circuit can be reconciled with 
each system. 

First, the number of simple irreducible circuits in any complete system 
must be tlie same for the same surface. 

Let a 1} ..., a p ; and b 1} ..., b n ; be two complete systems. Because a 1} ..., 
a p constitute a complete system, every circuit of the system of circuits b is 
reconcileable with it ; that is, integers ra# exist, such that 

b r = m lr a l + m. 2r a. 2 + . . . + m pr a p , 

for r = 1, 2, ..., n. If n were >p, then by combining linearly each equation 
after the first p equations with those p equations, and eliminating a l , ..., a p 
from the set of p + 1 equations, we could derive n p relations of the form 

M^ + M,b 2 + . . . + M n b n = 0, 

where the coefficients M, being determinants the constituents of which are 
integers, would be integers. The system of circuits b is irreducible, and there 
are therefore no such relations ; hence n is not greater than p. 



167.] ON CONNECTED SURFACES 329 

Similarly, by considering the reconciliation of each circuit a with the 
irreducible system of circuits b, it follows that p is not greater than n. 

Hence p and n are equal to one another. And, because each system is a 
complete system, there are integers A and B such that 
a r = A rl bi + A r2 b. 2 4- + A rn b n (r = I, ..., 
b s = Bg^ + B s . 2 a 2 + . . . -I- B m On (s = l, ..., 

The determinant of the integers A is equal to + 1 ; likewise the deter 
minant of the integers B. 

Secondly, let x be a circuit reconcileable with the system of circuits a : it is 
reconcileable with any other complete system of circuits. 

Since x is reconcileable with the system a, integers m 1} ..., m n can be 

found such that 

x = ??i 1 1 + . . . + m n a n . 

Any other complete system of n circuits b is such that the circuits a can 
be expressed in the form 

a r = Anbj. + ... + A rn b n , (r = 1, . . ., n), 
where the coefficients A are integers ; and therefore 

n n n 

x = b 1 2 m r A rl 4- 6 2 S m r A r z + . . . + b n X m r A rn 

r=l r=l r=l 

= g r i&i + g r 2&a + ~ +q n l>n, 

where the coefficients q are integers, that is, x is reconcileable with the 
complete system of circuits b. 

168. It thus appears that for the construction of any circuit on a surface, 
it is sufficient to know some one complete system of simple irreducible 
circuits. A complete system is supposed to contain the smallest possible 
number of simple circuits : any one which is reconcileable with the rest is 
omitted, so that the circuits of a system may be considered as independent. 
Such a system is indicated by the following theorems : 

I. No irreducible simple circuit can be drawn on a simply connected 
surface*. 

If possible, let an irreducible circuit G be drawn in a simply connected 
surface with a boundary B. Make a loop-cut along C, and change it into a 
cross-cut by making a cross-cut A from some point of C to a point of B ; 
this cross-cut divides the surface into two simply connected pieces, one of 
which is bounded by B, the two edges of A, and one edge of the cut along C, 
and the other of which is bounded entirely by the cut along C. 

The latter surface is smaller than the original surface ; it is simply 

connected and has a single boundary. If an irreducible simple circuit can 

be drawn on it, we proceed as before, and again obtain a still smaller simply 

connected surface. In this way, we ultimately obtain an infinitesimal 

* All surfaces considered are supposed to be bounded. 



330 RELATIONS BETWEEN CONNECTIVITY [168. 

element ; for every cut divides the surface, in which it is made, into 
distinct pieces. Irreducible circuits cannot be drawn in this element ; and 
therefore its boundary is reducible. This boundary is a circuit in a larger 
portion of the surface : the circuit is reducible so that, in that larger portion 
no irreducible circuit is possible and therefore its boundary is reducible. 
This boundary is a circuit in a still larger portion, and the circuit is 
reducible : so that in this still larger portion no irreducible circuit is possible 
and once more the boundary is reducible. 

Proceeding in this way, we find that no irreducible simple circuit is 
possible in the original surface. 

COROLLARY. No irreducible circuit can be drawn on a simply connected 
surface. 

II. A complete system of irreducible simple circuits for a surface of 
connectivity N contains N I simple circuits, so that every other circuit on the 
surface is reconcileable with that system. 

Let the surface be resolved by cross-cuts into a single simply connected 
surface: N 1 cross-cuts will be necessary. Let CD be 
any one of them : and let a and b be two points on the /e 

opposite edges of the cross-cut. Then since the surface is L n 

simply connected, a line can be drawn in the surface from 
a to b without passing out of the surface or without 
meeting a part of the boundary, that is, without meeting 
any other cross-cut. The cross-cut CD ends either in Fi s- 45 - 

another cross-cut or in a boundary; the line ae...fb 
surrounds that other cross-cut or that boundary as the case may be : hence, 
if the cut CD be obliterated, the line ae...fba is irreducible on the surface in 
which the other N 2 cross-cuts are made. But it meets none of those cross 
cuts; hence, when they are all obliterated so as to restore the unresolved 
surface of connectivity N, it is an irreducible circuit. It is evidently riot 
a repeated circuit; hence it is an irreducible simple circuit. Hence the 
line of an irreducible simple circuit on an unresolved surface is given by 
a line passing from a point on one edge of a cross-cut in the resolved 
surface to a point on the opposite edge. 

Since there are N -I cross-cuts, it follows that N 1 irreducible simple 
circuits can thus be obtained: one being derived in the foregoing manner 
from each of the cross-cuts, which are necessary to render the surface simply 
connected. It is easy to see that each of the irreducible circuits on an 
unresolved surface is, by the cross-cuts, rendered impossible as a circuit on 
the resolved surface. 

But every other irreducible circuit C is reconcileable with the Nl 
circuits, thus obtained. If there be one not reconcileable with these N-l 
circuits, then, when all the cross-cuts are made, the circuit C is not rendered 



168.] 



AND IRREDUCIBLE CIRCUITS 



331 



impossible, if it be not reconcileable with those which are rendered impossible 
by the cross-cuts : that is, there is on the resolved surface an irreducible 
circuit. But the resolved surface is simply connected, and therefore no 
irreducible circuit can be drawn on it : hence the hypothesis as to C, which 
leads to this result, is not tenable. 

Thus every other circuit is reconcileable with the system of N 1 circuits : 
and therefore the system is complete*. 

This method of derivation of the circuits at once indicates how far a 
system is arbitrary. Each system of cross-cuts leads to a complete system of 
irreducible simple circuits, and vice versa ; as the one system is not unique, 
so the other system is not unique. 

For the general question, Jordan s memoir, Des contours traces sur les surfaces, 
Liouville, 2 me Ser., t. xi., (1866), pp. 110130, may be consulted. 

Ex. 1. On a doubly connected surface, one irreducible simple circuit can be drawn. 
It is easily obtained by first resolving the surface into one that is simply connected 
a single cross-cut CD is effective for this purpose and then by drawing a curve aeb in the 




Fig. 46, (i). 

surface from one edge of the cross-cut to the other. All other irreducible circuits on the 
unresolved surface are reconcileable with the circuit aeba. 

Ex. 2. On a triply- connected surface, two independent irreducible circuits can be 




Fig. 46, (ii). 

* If the number of independent irreducible simple circuits be adopted as a basis for the 
definition of the connectivity of a surface, the result of the proposition would be taken as the 
definition : and the resolution of the surface into one, which is simply connected, would then be 
obtained by developing the preceding theory in the reverse order. 



332 



DEFORMATION 



[168. 



drawn. Thus in the figure C l and C 2 will form a complete system. The circuits C 3 and (7 4 
are also irreducible : they can evidently be deformed into C^ and <7 2 and reducible circuits 
by continuous deformation : in the algebraical notation adopted, we have 

C 3 =C 1 + C 2 , C i =C l -C. 2 . 

Ex. 3. Another example of a triply connected surface is given in Fig. 47. Two irredu 
cible simple circuits are C v and C%. Another irreducible circuit is C 3 ; this can be 




Fig. 47. 

reconciled with C l and C. 2 by drawing the point a into coincidence with the intersection 
of Cj and (7 2 , and the point c into coincidence with the same point. 

Ex. 4. As a last example, consider the surface of a solid sphere with n holes bored 
through it. The connectivity is 2n + 1 : hence 2n independent irreducible simple circuits 




Fig. 48. 

can be drawn on the surface. The simplest complete system is obtained by taking 2n 
curves : made up of a set of n, each round one hole, and another set of n, each through 
one hole. 

A resolution of this surface is given by taking cross-cuts, one round each hole (making 
the circuits through the holes no longer possible) and one through each hole (making the 
circuits round the holes no longer possible). 

The simplest case is that for which n= 1 : the surface is equivalent to the anchor-ring. 

169. Surfaces are at present being considered in view of their use as a 
means of representing the value of a complex variable. The foregoing inves 
tigations imply that surfaces can be classed according to their connectivity ; 
and thus, having regard to their designed use, the question arises as to 
whether all surfaces of the same connectivity arc equivalent to one another, 
so as to be transformable into one another. 



169.] OF CONNECTED SURFACES 333 

Moreover, a surface can be physically deformed and still remain suitable for 
representation of the variable, provided certain conditions are satisfied. We 
thus consider geometrical transformation as well as physical deformation ; but 
we are dealing only with the general results and not with the mathematical 
relations of stretching and bending, which are discussed in treatises on 
Analytical Geometry*. 

It is evident that continuity is necessary for both : discontinuity would 
imply discontinuity in the representation of the variable. Points that are 
contiguous (that is, separated only by small distances measured in the surface) 
must remain contiguous -f*: and one point in the unchanged surface must 
correspond to only one point in the changed surface. Hence in the continuous 
deformation of a surface there may be stretching and there may be bending ; 
but there must be no tearing and there must be no joining. 

For instance, a single untwisted ribbon, if cut, comes to be simply connected. If a twist 
through 180 be then given to one end and that end be then joined to the other, we shall 
have a once- twisted ribbon, which is a surface with only one face and only one edge; 
it cannot be looked upon as an equivalent of the former surface. 

A spherical surface with a single hole can have the hole stretched and the surface 
flattened, so as to be the same as a bounded portion of a plane : the two surfaces are 
equivalent to one another. Again, in the spherical surface, let a large indentation be 
made : let both the outer and the inner surfaces be made spherical ; and let the mouth of 
the indentation be contracted into the form of a long, narrow hole along a part of a great 
circle. When each point of the inner surface is geometrically moved so that it occupies the 
position of its reflexion in the diametral plane of the hole, the final form of the whole 
surface is that of a two-sheeted surface with a junction along a line : it is a spherical 
winding-surface, and is equivalent to the simply connected spherical surface. 

170. It is sufficient, for the purpose of representation, that the two 
surfaces should have a point-to-point transformation : it is not necessary 
that physical deformation, without tears or joins, should be actually possible. 
Thus a ribbon with an even number of twists would be as effective as a 
limited portion of a cylinder, or (what is the same thing) an untwisted ribbon : 
but it is not possible to deform the one into the other physically |. 

It is easy to see that either deformation or transformation of the kind 
considered will change a bifacial surface into a bifacial surface ; that it will 
not alter the connectivity, for it will not change irreducible circuits into 

* See, for instance, Frost s Solid Geometry, (3rd ed.), pp. 342 352. 

t Distances between points must be measured along the surface, not through space ; the 
distance between two points is a length which one point would traverse before reaching the 
position of the other, the motion of the point being restricted to take place in the surface. 
Examples will arise later, in Biemann s surfaces, in which points that are contiguous in space 
are separated by finite distances on the surface. 

Clifford, Coll. Hath. Papers, p. 250. 

J The difference between the two cases is that, in physical deformation, the surfaces are the 
surfaces of continuous matter and are impenetrable ; while, in geometrical transformation, the 
surfaces may be regarded as penetrable without interference with the continuity. 



334 DEFORMATION OF SURFACES [170. 

reducible circuits, and the number of independent irreducible circuits 
determines the connectivity: and that it will not alter the number of boundary 
curves, for a boundary will be changed into a boundary. These are necessary 
relations between the two forms of the surface : it is not difficult to see that 
they are sufficient for correspondence. For if, on each of two bifacial surfaces 
with the same number of boundaries and of the same connectivity, a complete 
system of simple irreducible circuits be drawn, then, when the members of the 
systems are made to correspond in pairs, the full transformation can be effected 
by continuous deformation of those corresponding irreducible circuits. It 
therefore follows that : 

The necessary and sufficient conditions, that two bifacial surfaces may be 
equivalent to one another for the representation of a variable, are that tlie two 
surfaces should be of the same connectivity and should have the same number of 
boundaries. 

As already indicated, this equivalence is a geometrical equivalence : 
deformation may be (but is not of necessity) physically possible. 

Similarly, the presence of one or of several knots in a surface makes no 
essential difference in the use of the surface for representing a variable. Thus 
a long cylindrical surface is changed into an anchor-ring when its ends are 
joined together ; but the changed surface would be equally effective for 
purposes of representation if a knot were tied in the cylindrical surface before 
the ends are joined. 

But it need hardly be pointed out that though surfaces, thus twisted or 
knotted, are equivalent for the purpose indicated, they are not equivalent for 
all topological enumerations. 

Seeing that bifacial surfaces, with the same connectivity and the same 
number of boundaries, are equivalent to one another, it is natural to adopt, as 
the surface of reference, some simple surface with those characteristics; thus 
for a surface of connectivity 2p + 1 with a single boundary, the surface of a 
solid sphere, bounded by a point and pierced through with p holes, could be 
adopted. 

Klein calls* such a surface of reference a Normal Surface. 

It has been seen that a bounded spherical surface and a bounded simply connected 
part of a plane are equivalent they are, moreover, physically deformable into one 
another. 

An untwisted closed ribbon is equivalent to a bounded piece of a plane with one hole 
in it they are deformable into one another : but if the ribbon, previous to being closed, 
have undergone an even number of twists each through 180, they are still equivalent 
but are not physically deformable into one another. Each of the bifacial surfaces is 
doubly connected (for a single cross-cut renders each simply connected) and each of them 

* Ueber Riemann s Theorie der algebraischen Functionen und ihrer Integrate, (Leipzig, 
Teubner, 1882), p. 26. 



170.] REFERENCES 335 

has two boundaries. If however the ribbon, previous to being closed, have imdcrgone 
an odd number of twists each through 180, the surface thus obtained is not equivalent to 
the single-holed portion of the plane ; it is unifacial arid has only one boundary. 

A spherical surface pierced in n-\-l holes is equivalent to a bounded portion of the 
plane with n holes ; each is of connectivity n + 1 and has n + 1 boundaries. The spherical 
surface can be deformed into the plane surface by stretching one of its holes into the form 
of the outside boundary of the plane surface. 

Ex. Prove that the surface of a bounded anchor-ring can be physically deformed into 
the surface in Fig. 47, p. 332. 



For continuation and fuller development of the subjects of the present chapter, the 
following references, in addition to those which have been given, will be found useful : 

Klein, Math. Ann., t. vii, (1874), pp. 548557; ib., t. ix, (1876), pp. 476482. 

Lippich, Math. Ann., i. vii, (1874), pp. 212 229 ; Wiener Sitzungsb., t. Ixix, (ii), 
(1874), pp. 9199. 

Durege, Wiener Sitzungsb., t. Ixix, (ii), (1874), pp. 115120; and section 9 of his 
treatise, quoted on p. 316, note. 

Neumann, chapter vii of his treatise, quoted on p. 5, note. 

Dyck, Math. Ann., t. xxxii, (1888), pp. 457512, ib., t. xxxvii, (1890), pp. 273316; 

at the beginning of the first part of this investigation, a valuable series of references 
is given. 

Dingeldey, Topologische Studien, (Leipzig, Teubner, 1890). 



CHAPTER XV. 

RIEMANN S SURFACES. 

171. THE method of representing a variable by assigning to it a position 
in a plane or on a sphere is effective when properties of uniform functions of 
that variable are discussed. But when multiform functions, or integrals of 
uniform functions occur, the method is effective only when certain parts of 
the plane are excluded, due account being subsequently taken of the effect of 
such exclusions; and this process, the extension of Cauchy s method, was 
adopted in Chapter IX. 

There is another method, referred to in 100 as due to Riemann, of an 
entirely different character. In Riemann s representation, the region, in 
which the variable z exists, no longer consists of a single plane but of a 
number of planes ; they are distinct from one another in geometrical concep 
tion, yet, in order to preserve a representation in which the value of the 
variable is obvious on inspection, the planes are infinitesimally close to one 
another. The number of planes, often called sheets, is the same as the 
number of distinct values (or branches) of the function w for a general 
argument z and, unless otherwise stated, will be assumed finite; each sheet 
is associated with one branch of the function, and changes from one branch 
of the function to another are effected by making the ^-variable change 
from one sheet to another, so that, to secure the possibility of change 
of sheet, it is necessary to have means of passage from one sheet to another. 
The aggregate of all the sheets is a surface, often called a Riemanns 
Surface. 

For example, consider the function 

w=z* + (z-I}~*, 

the cube roots being independent of one another. It is evidently a nine-valued function ; 
the number of sheets in the appropriate Eiemann s surface is therefore nine. 

The branch-points are 2 = 0, z = l, 2=00. Let o> and a denote a cube-root of unity, 
independently of one another ; then the values of z* can be represented in the form 



171.] 



EXAMPLES OF RIEMANN s SURFACES 



337 



ill -A - 4 

2 3 , C02 3 ", co 2 2*; and the values of (2-!) 3 can be represented in the form (2-!) , 

^(z - \ ) ~ 3 } ( - 1) The nine values of w can be symbolically expressed as follows : 



Fig. 49. 



Fig. 50. 



where the symbols opposite to w give the coefficients of z 3 and of (2- 1) 3 respectively. 

Now when 2 describes a small simple circuit positively round the origin, the groups 
in cyclical order are u\, w 2 , w 3 ; w 4 , w 5 , w 6 ; w r , w 8 , io 9 . And therefore, in the immediate 
vicinity of the origin, there must be means of passage to enable 
the 2-point to make the corresponding changes from sheet to 
sheet. Taking a section of the whole surface near the origin ~ 
so as to indicate the passages and regarding the right-hand 
sides as the part from which the 2-variable moves when it 
describes a circuit positively, the passages must be in character as 
indicated in Fig. 49. And it is evident that the further descrip 
tion of small simple circuits round the origin will, with these passages, lead to the proper 
values : thus %, which after the single description is the value of w 4 , becomes w 6 after 
another description and it is evident that a point in the w- sheet passes into the w 6 sheet. 

When 2 describes a small simple circuit positively round the point 1, the groups in cyclical 
order are w lt ^ 4 , %; w 2 , w 5 , w s ; w 3 , w 6 , w 9 : and therefore, 
in the immediate vicinity of the point 1, there must be ~ 
means of passage to render possible the corresponding changes 
of 2 from sheet to sheet. Taking a section as before near the ~ 
point 1 and with similar convention as to the positive direc 
tion of the 2-path, the passages must be in character as 
indicated in Fig. 50. 

Similarly for infinitely large values of 2. 

If then the sheets can be so joined as to give these possibilities of passage and also 
give combinations of them corresponding to combinations of the simple paths indicated, 
then there will be a surface to any point of which will correspond one and only one value 
of w : and when the value of w is given for a point 2 in an ordinary plane of variation, 
then that value of w will determine the sheet of the surface in which the point 2 is to 
be taken. A surface will then have been constructed such that the function w, which is 
multiform for the single-plane representation of the variable, is uniform for variations 
in the many-sheeted surface. 

Again, for the simple example arising from the two-valued function, defined by 
the equation 

w = {(z-a}(z-b}(z-c}}-\ 

the branch-points are a, b, c, oo ; and a small simple circuit round any one of these 
four points interchanges the two values. The Riemann s surface is two-sheeted and 
there must be means of passage between the two sheets in the vicinity of a, that of b, 
that of c and at the infinite part of the plane. 

These examples are sufficient to indicate the main problem. It is the 
construction of a surface in which the independent variable can move so 
F. 22 



338 SHEETS OF HIEMANN S SURFACE [171. 

that, for variations of z in that surface, the multiformity of the function is 
changed to uniformity. From the nature of the case, the character of the 
surface will depend on the character of the function : and thus, though all the 
functions are uniform within their appropriate surfaces, these surfaces are 
widely various. Evidently for uniform functions of z the appropriate surface 
on the above method is the single plane already adopted. 

172. The simplest classes of functions for which a Riemaim s surface is 
useful are (i) those called ( 94) algebraic functions, that is, multiform functions 
of the independent variable denned by an algebraical equation of the form 



which is of finite degree, say n, in w, and (ii) those usually called Abelian 
functions, which arise through integrals connected with algebraic functions. 

Of such an algebraic function there are, in general, n distinct values ; but 
for the special values of z, that are the branch-points, two or more of the 
values coincide. The appropriate Riemann s surface is composed of n sheets ; 
one branch, and only one branch, of w is associated with a sheet. The 
variable z, in its relation to the function, is determined not merely by its 
modulus and argument but also by its sheet ; that is, in the language of the 
earlier method, we take account of the path by which z acquires a value. The 
particular sheet in which z lies determines the particular branch of the 
function. Variations of #, which occur within a sheet and do not coincide 
with points lying in regions of passage between the sheets, lead to variations 
in the value of the branch of w associated with the sheet ; a return to an 
initial value of z, by a path that nowhere lies within a region of passage, 
leaves the ^-point in the same sheet as at first and so leads to the initial 
branch (and to the initial value of the branch) of w. But a return to an 
initial value of z by a path, which, in the former method of representation, 
would enclose a branch-point, implies a change of the branch of the function 
according to the definite order prescribed by the branch-point. Hence the 
final value of the variable z on the Riemann s surface must lie in a sheet that 
is different from that of the initial (and algebraically equal) value ; and 
therefore the sheets must be so connected that, in the immediate vicinity of 
branch-points, there are means of passage from one sheet to another, securing 
the proper interchanges of the branches of the function as defined by the 
equation. 

173. The first necessity is therefore the consideration of the mode in 
which the sheets of a Riemann s surface are joined : the mode is indicated by 
the theorem that sheets of a Riemann s surface are joined along lines. 

The junction might be made either at a point, as with two spheres in 
contact, or by a common portion of a surface, as with one prism lying on 



173.] JOINED ALONG BRANCH-LINES 339 

another, or along lines ; but whatever the character of the junction be, it 
must be such that a single passage across it (thereby implying entrance to 
the junction and exit from it) must change the sheet of the variable. 

If the junction were at a point, then the - variable could change from one 
sheet into another sheet, only if its path passed through that point : any 
other closed path would leave the z- variable in its original sheet. A small 
closed curve, infinites! rn ally near the point and enclosing it and no other 
branch-point, is one which ought to transfer the variable to another sheet 
because it encloses a branch-point : and this is impossible with a point-junction 
when the path does not pass through the point. Hence a junction at a point 
only is insufficient to provide the proper means of passage from sheet to 
sheet. 

If the junction were effected by a common portion 
of surface, then a passage through it (implying an 
entrance into that portion and an exit from it) ought to 
change the sheet. But, in such a case, closed contours .- -- 
can be constructed which make such a passage without Fi 8- 51> 

enclosing the branch-point a : thus the junction would cause a change of 
sheet for certain circuits the description of which ought to leave the 
z- variable in the original sheet. Hence a junction by a continuous area of 
surface does not provide proper means of passage from sheet to sheet. 

The only possible junction which remains is a line. 

The objection in the last case does not apply to a closed / ^ 

contour which does not contain the branch-point ; for the /.--" 

line cuts the curve twice and there are therefore two Fig. 52. 

crossings ; the second of them makes the variable return to the sheet which 
the first crossing compelled it to leave. 

Hence the junction between any two sheets takes place along a line. 

Such a line is called* a branch-line. The branch -points of a multiform 
function lie on the branch-lines, after the foregoing explanations ; and a 
branch-line can be crossed by the variable only if the variable change its 
sheet at crossing, in the sequence prescribed by the branch-point of the 
function which lies on the line. Also, the sequence is reversed when the 
branch-line is crossed in the reversed direction. 

Thus, if two sheets of a surface be connected along a branch-line, a point which 
crosses the line from the first sheet must pass into the second and a point which crosses 
the line from the second sheet must pass into the first. 

Again, if, along a common direction of branch-line, the first sheet of a surface 
be connected with the second, the second with the third, and the third with 

* Sometimes cross-line, sometimes branch-section. The German title is Verzweigungschnitt; 
the French is lignc de passage ; see also the note on the equivalents of branch-point, p. 15. 

222 



340 PROPERTIES OF BRANCH-LINES [173. 

the first, a point which crosses the line from the first sheet in one direction must pass 
into the second sheet, but if it cross the line in the other direction it must pass into 
the third sheet. 

A branch -point does not necessarily affect all the branches of a function : 
when it affects only some of them, the corresponding property of the Riemann s 
surface is in evidence as follows. Let z=a determine a branch-point affecting, 
say, only r branches. Take n points a, one in each of the sheets ; and through 
them draw n lines cab, having the same geometrical position in the respective 
sheets. Then in the vicinity of the point a in each of the n sheets, associated 
with the r affected branches, there must be means of passage from each one 
to all the rest of them ; and the lines cab can conceivably be the branch-lines 
with a properly established sequence. The point a does not affect the other 
n r branches : there is therefore no necessity for means of passage in the 
vicinity of a among the remaining n r sheets. In each of these remaining 
sheets, the point a and the line cab belong to their respective sheets alone : 
for them, the point a is not a branch-point and the line cab is not a branch- 
line. 

174. Several essential properties of the branch-lines are immediate 
inferences from these conditions. 

I. A free end of a branch-line in a surface is a branch-point. 

Let a simple circuit be drawn round the free end so small as to enclose no 
branch-point (except the free end, if it be a branch-point). The circuit meets 
the branch-line once, and the sheet is changed because the branch-line is 
crossed ; hence the circuit includes a branch-point which therefore can be 
only the free end of the line. 

Note. A branch-line may terminate in the boundary of the surface, 
and then the extremity need not be a branch-point. 

II. When a branch-line extends beyond a branch-point lying in its course, 
the sequence of interchange is not the same on the two sides of the point. 

If the sequence of interchange be the same on the two sides of the branch 
point, a small circuit round the point would first cross one part of the branch- 
line and therefore involve a change of sheet and then, in its course, would 
cross the other part of the branch-line in the other direction which, on the 
supposition of unaltered sequence, would cause a return to the initial sheet. 
In that case, a circuit round the branch-point would fail to secure the proper 
change of sheet. Hence the sequence of interchange caused by the branch- 
line cannot be the same on the two sides of the point. 

III. If two branch-lines with different sequences of interchange have a 
common extremity, that point is either a branch-point or an extremity of at 
least one other branch-line. 




174.] SYSTEM OF BRANCH-LINES 341 

If the point be not a branch-point, then a simple curve enclosing it, taken 
so small as to include no branch-point, must 
leave the variable in its initial sheet. Let A 
be such a point, AB and AC be two branch- 
lines having A for a common extremity ; let ., A ,. ^ 

the sequence be as in the figure, taken for a F . 

simple case ; and suppose that the variable 

initially is in the rth sheet. A passage across AB makes the variable 
pass into the sth sheet. If there be no branch-line between AB and AC 
having an extremity at A, and if neither n nor m be s, then the passage 
across AC makes no change in the sheet of the variable and, therefore, in 
order to restore r before AB, at least one branch-line must lie in the angle 
between AC and AB, estimated in the positive trigonometrical sense. 

If either n or m, say n, be s, then after passage across AC, the point is in 
the mt\i sheet ; then, since the sequences are not the same, m is not r and 
there must be some branch-line between AC and AB to make the point 
return to the rth sheet on the completion of the circuit. 

If then the point A be not a branch-point, there must be at least one 
other branch-line having its extremity at A. This proves the proposition. 

COROLLARY 1. If both of two branch-lines extend beyond a point of inter 
section, which is not a branch-point, no sheet of the surface has both of them for 
branch-lines. 

COROLLARY 2. If a change of sequence occur at any point of a branch- 
line, then either that point is a branch-point or it lies also on some other 
branch-line. 

COROLLARY 3. No part of a branch-line with only one branch-point on it 
can be a closed curve. 

It is evidently superfluous to have a branch-line without any branch-point 
on it. 

175. On the basis of these properties, we can obtain a system of branch- 
lines satisfying the requisite conditions which are : 

(i) the proper sequences of change from sheet to sheet must be 
secured by a description of a simple circuit round a branch 
point : if this be satisfied for each of the branch-points, it 
will evidently be satisfied for any combination of simple circuits, 
that is, for any path whatever enclosing one or more branch 
points. 

(ii) the sheet, in which the variable re-assumes its initial value after 
describing a circuit that encloses no branch-point, must be the 
initial sheet. 



342 SYSTEM OF BRANCH-LINES [175. 

In the ^-plane of Cauchy s method, let lines be drawn from any point I, not 
a branch-point in the first instance, to each of the branch-points, as in fig. 19, 
p. 156, so that the joining lines do not meet except at /: and suppose the 
w-sheeted Riemann s surface to have branch-lines coinciding geometrically 
with these lines, as in 173, and having the sequence of interchange for 
passage across each the same as the order in the cycle of functional values 
for a small circuit round the branch-point at its free end. No line (or part 
of a line) can be a closed curve ; the lines need not be straight, but they 
will be supposed drawn as direct as possible to the points in angular 
succession. 

The first of the above requisite conditions is satisfied by the establish 
ment of the sequence of interchange. 

To consider the second of the conditions, it is convenient to divide 
circuits into two kinds, (a) those which exclude /, (/3) those which include /, 
no one of either kind (for our present purpose) including a branch-point. 

A closed circuit, excluding I and all the branch-points, must intersect a 
branch-line an even number of times, if 
it intersect the line in real points. Let 
the figure (fig. 54) represent such a case : 
then the crossings at A and B counter 
act one another and so the part be 
tween A and B may without effect be 
transferred across IB 3 so as not to cut 
the branch-line at all. Similarly for 
the points C and D : and a similar 
transference of the part now between 
C and D may be made across the 
branch-line without effect: that is, the 
circuit can, without effect, be changed 
so as not to cut the branch-line IB S at all. A similar change can be made 
for each of the branch-lines : and so the circuit can, without effect, be changed 
into one which meets no branch-line and therefore, on its completion, leaves 
the sheet unchanged. 

A closed circuit, including / but no branch-point, must meet each branch- 
line an odd number of times. A change similar in character to that in 
the previous case may be made for each branch-line : and without affecting 
the result, the circuit can be changed so that it meets each branch-line only 
once. Now the effect produced by a branch-line on the function is the same 
as the description of a simple loop round the branch-point which with / 
determines the branch-line : and therefore the effect of the circuit at present 
contemplated is, after the transformation which does not affect the result, the 
same as that of a circuit, in the previously adopted mode of representation, 




175.] FOR A SURFACE 

enclosing all the branch-points. But, by Cor. III. of 90, the effect of a 
circuit which encloses all the branch-points (including z = GO , if it be a 
branch-point) is to restore the value of the function which it had at the 
beginning of the circuit : and therefore in the present case the effect is to 
make the point return to the sheet in which it lay initially. 

It follows therefore that, for both kinds of a closed circuit containing no 
branch-point, the effect is to make the ^-variable return to its initial sheet 
on resuming its initial value at the close of the circuit. 

Next, let the point / be a branch-point ; and let it be joined by lines, 
as direct* as possible, to each of the other branch -points in angular succes 
sion. These lines will be regarded as the branch-lines ; and the sequence of 
interchange for passage across any one is made that of the interchange pre 
scribed by the branch-point at its free extremity. 

The proper sequence of change is secured for a description of a simple 
closed circuit round each of the branch-points other than /. Let a small 
circuit be described round /; it meets each of the branch-lines once and 
therefore its effect is the same as, in the language of the earlier method of 
representing variation of z, that of a circuit enclosing all the branch-points 
except 7. Such a circuit, when taken on the Neumann s sphere, as in Cor. 
III., 90 and Ex. 2, 104, may be regarded in two ways, according as one or 
other of the portions, into which it divides the area of the sphere, is regarded 
as the included area; in one way, it is a circuit enclosing all the branch 
points except /, in the other it is a circuit enclosing / alone and no other 
branch-point. Without making any modification in the final value of w, it 
can (by 90) be deformed, either into a succession of loops round all the 
branch-points save one, or into a loop round that one ; the effect of these two 
deformations is therefore the same. Hence the effect of the small closed 
circuit round / meeting all the branch-lines is the same as, in the other mode 
of representation, that of a small curve round / enclosing no other branch 
point ; and therefore the adopted set of branch- lines secures the proper 
sequence of change of value for description of a circuit round /. 

The first of the two necessary conditions is therefore satisfied by the 
present arrangement of branch-lines. 

The proof, that the second of the two necessary conditions is also satisfied 
by the present arrangement of branch-lines, is similar to that in the preceding 
case, save that only the first kind of circuit of the earlier proof is possible. 

Jt thus appears that a system of branch-lines can be obtained which 
secures the proper changes of sheet for a multiform function : and therefore 
Riemann s surfaces can be constructed for such a function, the essential 
property being that over its appropriate surface an otherwise multiform 
function of the variable is a uniform function. 

* The reason for this will appear in 183, 184. 



344 EXAMPLES [175. 

The multipartite character of the function has its influence preserved by 
the character of the surface to which the function is referred : the surface, 
consisting of a number of sheets joined to one another, may be a multiply 
connected surface. 

In thus proving the general existence of appropriate surfaces, there has 
remained a large arbitrary element in their actual construction : moreover, 
in particular cases, there are methods of obtaining varied configurations of 
branch-lines. Thus the assignment of the n branches to the n sheets has 
been left unspecified, and is therefore so far arbitrary : the point I, if not a 
branch-point, is arbitrarily chosen and so there is a certain arbitrariness of 
position in the branch -lines. Naturally, what is desired is the simplest 
appropriate surface : the particularisation of the preceding arbitrary qualities 
is used to derive a canonical form of the surface. 

176. The discussion of one or two simple cases will help to illustrate the 
mode of junction between the sheets, made by branch-lines. 

The simplest case of all is that in which the surface has only a single 
sheet: it does not require discussion. 

The case next in simplicity is that in which the surface is two-sheeted : 
the function is therefore two- valued and is consequently defined by a 
quadratic equation of the form 

Lu a + 2Mu + N = 0, 

where L and M are uniform functions of z. When a new variable w is 
introduced, defined by Lu + M=w, so that values of iv and of u correspond 
uniquely, the equation is 



It is evident that every branch-point of u is a branch-point of w, and 
vice versa ; hence the Riemann s surface is the same for the two equations. 
Now any root of P (z) of odd degree is a branch-point of iv. If then 



where R (z} is a product of only simple factors, every factor of R (z) leads to 
a branch-point. If the degree of R (z} be even, the number of branch-points 
for finite values of the variable is even and z = oo is not a branch-point ; if the 
degree of R(z) be odd, the number of branch -points for finite values of the 
variable is odd and z = oo is a branch-point : in either case, the number of 
branch-points is even. 

There are only two values of w, and the Riemann s surface is two-sheeted: 
crossing a branch-line therefore merely causes a change of sheet. The free 
ends of branch-lines are branch-points ; a small circuit round any branch 
point causes an interchange of the branches w, and a circuit round any two 
branch-points restores the initial value of w at the end and therefore leaves 
the variable in the same sheet as at the beginning. These are the essential 
requirements in the present case ; all of them are satisfied by taking each 



176.] OF RIEMANN S SURFACES 345 

branch-line as a line connecting two (and only two) of the branch-points. The 
ends of all the branch -lines are free : and their number, in this method, is 
one-half that of the (even) number of branch-points. A small circuit round 
a branch-point meets a branch-line once and causes a change of sheet ; a 
circuit round two (and not more than two) branch -points causes either no 
crossing of branch-line or an even number of crossings and therefore restores 
the variable to the initial sheet. 

A branch-line is, in this case, usually drawn in the form of a straight line 
when the surface is plane : but this form is not essential and all that is 
desirable is to prevent intersections of the branch-lines. 

Note. Junction between the sheets along a branch-line is easily secured. 
The two sheets to be joined are cut along the branch-line. One edge of the 
cut in the upper sheet, say its right edge looking along the section, is joined 
to the left edge of the cut in the lower sheet ; and the left edge in the upper 
sheet is joined to the right edge in the lower. 

A few simple examples will illustrate these remarks as to the sheets : illustrations of 
closed circuits will arise later, in the consideration of integrals of multiform functions. 

Ex. 1. Let w* = A(z-a)(z-b}, 

so that a and b are the only branch-points. The surface is two-sheeted : the line ab may 
be made the branch-line. In Fig. 55 only part of the upper sheet is shewn*, as likewise 
only part of the lower sheet. Continuous lines imply what is visible ; arid dotted lines 
what is invisible, on the supposition that the sheets are opaque. 

The circuit, closed in the surface and passing round 0, is made up of OJK in the upper 
sheet : the point crosses the branch-line and then passes into the lower sheet, where it 
describes the dotted line KLH : it then meets and crosses the branch-line at If, passes 
into the upper sheet and in that sheet returns to 0. Similarly of the line ABC, the part 
AB lies in the lower sheet, the part EC in the upper : of the line DG the part DE lies in 
the upper sheet, the part EFG in the lower, the piece FG of this part being there visible 
beyond the boundary of the retained portion of the upper surface. 

Ex. 2. Let Aw? 2 = z 3 -a 3 . 

The branch-points (Fig. 56) are A ( = a), B ( = ), (7( = aa 2 ), where a is a primitive cube 
root of unity, and 2 = 00. The branch -lines can be made by BC, Ace ; and the two- 
sheeted surface will be a surface over which w is uniform. Only a part of each sheet 
is shewn in the figure; a section also is made at M across the surface, cutting the branch - 
line A QO . 

Ex. 3. Let w m =z n , 

where n and TO are prime to each other. The branch-points are z = and 2=00 ; and the 
branch-line extends from to QO . There are m sheets ; if we associate them in order with 
the branches w s , where 



w a =re 

for s=l, 2, ..., TO, then the first sheet is connected with the second forwards, the second 
with the third forwards, and so on ; the mth being connected with the first forwards. 

* The form of the three figures in the plate opposite p. 346 is suggested by Holzmiiller, Ein- 
fiihrung in die Theorie der isogonalen Vericandschaften und der confomien AbbUdimgen, (Leipzig, 
Teubner, 1882), in which several illustrations are given. 



346 



SPHERICAL RIEMANN S SURFACE 



[176. 



The surface is sometimes also called a winding-surface; and a branch-point such as 
z0 on the surface, where a number m of sheets pass into one another in succession, is 
also called a winding-point of order m 1 (see p. 15, note). An illustration of the surface 
for m = 3 is given in Fig. 57, the branch-line being cut so as to shew the branching : what 
is visible is indicated by continuous lines ; what is in the second sheet, but is invisible, is 
indicated by the thickly dotted line ; what is in the third sheet, but is invisible, is indic 
ated by the thinly dotted line. 

Ex. 4. Consider a three-sheeted surface having four branch-points at a, b, c, d ; and 
let each point interchange two branches, say, w. 2 , w 3 at a ; iv^ w 3 ai b ; w 2 , w 3 at c ; w lt w 2 






at d ; the points being as in Fig. 58. It is easy to verify that these branch-points 
satisfy the condition that a circuit, enclosing them all, restores the initial value of w. 

The branching of the sheets may be made as in the figure, the integers on the two sides 
of the line indicating the sheets that are to be joined along the line. 

A canonical form for such a surface can be derived from the more general case given 
later (in 186189). 

Ex. 5. Shew that, if the equation 



be of degree n in w and be irreducible, all the n sheets of the surface are connected, that 
is, it is possible by an appropriate path to pass from any sheet to any other sheet. 

177. It is not necessary to limit the surface representing the variable to 
a set of planes; and, indeed, as with uniform functions, there is a convenience 
in using the sphere for the purpose. 

We take n spheres, each of diameter unity, touching the Riemann s plane 
surface at a point A ; each sphere is regarded as the stereographic projection 
of a plane sheet, with regard to the other extremity A of the spherical 
diameter through A. Then, the sequence of these spherical sheets being 
the same as the sequence of the plane sheets, branch-points in the plane 
surface project into branch-points on the spherical surface : branch -lines be 
tween the plane sheets project into branch-lines between the spherical sheets 
and are terminated by corresponding points ; and if a branch-line extend in 
the plane surface to z=co, the corresponding branch-line in the spherical 
surface is terminated at A . 

A surface will thus be obtained consisting of n spherical sheets; like 
the plane Riemann s surface, it is one over which the n-valued function is a 
uniform function of the position of the variable point. 




Fig. 




M =-00 




To face p. 346. 



Fig. 57. 



177.] CONNECTIVITY OF A RIEMANN s SURFACE 347 

But also the connectivity of the n-sheeted spherical surface is the same as 
that of the n-sheeted plane surface with which it is associated. 

In fact, the plane surface can be mechanically changed into the spherical 
surface without tearing, or repairing, or any change except bending and 
compression: all that needs to be done is that the n plane sheets shall be 
bent, without making any change in their sequence, each into a spherical 
form, and that the boundaries at infinity (if any) in the plane-sheet shall 
be compressed into an infinitesimal point, being the South pole of the cor 
responding spherical sheet or sheets. Any junctions between the plane 
sheets extending to infinity are junctions terminated at the South pole. As 
the plane surface has a boundary, which, if at infinity on one of the sheets, is 
therefore not a branch-line for that sheet, so the spherical surface has a 
boundary which, if at the South pole, cannot be the extremity of a branch- 
line. 

178. We proceed to obtain the connectivity of a Riemann s surface : it 
is determined by the following theorem : 

Let the total number of branch-points in a Riemann s n-sheeted surface be 
r ; and let the number of, branches of the function interchanging at the first 
point be m l , the number interchanging at the second be m. 2 , and so on. Then 
the connectivity of the surface is 

fl-2n + 3, 
where fl denotes m, 1 + m 2 + ... + m r r. 

Take* the surface in the bounded spherical form, the connectivity N of 
which is the same as that of the plane surface : and let the boundary be a 
small hole A in the outer sheet. By means of cross-cuts and loop-cuts, the 
surface can be resolved into a number of distinct simply connected pieces. 

First, make a slice bodily through the sphere, the edge in the 
outside sheet meeting A and the direction of the 
slice through A being chosen so that none of the 
branch -points lie in any of the pieces cut off. Then n 
parts, one from each sheet and each simply connected, 
are taken away. The remainder of the surface has a 
cup-like form ; let the connectivity of this remainder 
be M. 

This slice has implied a number of cuts. 

The cut made in the outside sheet is a cross-cut, 
because it begins and ends in the boundary A. It 
divides the surface into two distinct pieces, one being 
the portion of the outside sheet cut off, and this piece is simply connected ; 

* The proof is founded on Neumann s, pp. 108 172. 




348 CONNECTIVITY OF A SURFACE [178. 

hence, by Prop. III. of 160, the remainder has its connectivity still repre 
sented by N. 

The cuts in all the other sheets, caused by the slice, are all loop-cuts, 
because they do not anywhere meet the boundary. There are n 1 loop- 
cuts, and each cuts off a simply connected piece ; and the remaining surface 
is of connectivity M. Hence, by Prop. V. of 161, 

M + n - 1 = N + 2 (n - 1), 
and therefore M = N+nl. 

In this remainder, of connectivity M, make r 1 cuts, each of which 
begins in the rim and returns to the rim, and is to be made through the n 
sheets together ; and choose the directions of these cuts so that each of the 
r resulting portions of the surface contains one (and only one) of the branch 
points. 

Consider the portion of the surface which contains the branch-point 
where m l sheets of the surface are connected. The m l connected sheets 
constitute a piece of a winding-surface round the winding-point of order 
m l 1 ; the remaining sheets are unaffected by the winding-point, and 
therefore the parts of them are n m^ distinct simply connected pieces. 
The piece of winding- surface is simply connected ; because a circuit, that 
does not contain the winding-point, is reducible without passing over the 
winding-point, and a circuit, that does contain the winding-point, is reducible 
to the winding-point, so that no irreducible circuit can be drawn. Hence 
the portion of the surface under consideration consists of n m l + 1 distinct 
simply connected pieces. 

Similarly for the other portions. Hence the total number of distinct 
simply connected pieces is 

r 

2 (n - m q + 1) 

9 = 1 

r 

= m 2 m q + r 
l-i 

= nr fl. 

But in the portion of connectivity M each of the r 1 cuts causes, in 
each of the sheets, a cut passing from the boundary and returning to the 
boundary, that is, a cross-cut. Hence there are n cross-cuts from each of the 
r\ cuts, and therefore n (r 1) cross-cuts altogether, made in the portion of 
surface of connectivity M. 

The effect of these n(r 1) cross-cuts is to resolve the portion of con 
nectivity M into nr l distinct simply connected pieces ; hence, by 160, 

M = n (r - 1) - (nr - H) + 2, 

and therefore N = M (n 1) = n - 2n + 3, 

the connectivity of the Riemann s surface. 



178.] CLASS OF A SURFACE 349 

r 

The quantity H, having the value 2 (m q 1), may be called the rami- 

</=i 

fication of the surface, as indicating the aggregate sum of the orders of 
the different branch-points. 

Note. The surface just considered is a closed surface to which a point 
has been assigned for boundary; hence, by Cor. I., Prop. III., 164, its 
connectivity is an odd integer. Let it be denoted by 2p + 1 ; then 

2p = ft - 2/i + 2, 

and 2p is the number of cross-cuts which change the Riemann s surface into 
one that is simply connected. 

The integer p is often called (Cor. I., Prop. III., 164) the class of the 
Riemann s surface; and the equation 

f(w, z) = 

is said to be of class p, when p is the class of the associated Riemann s 
surface. 

Ex. 1. When the equation is 

w> = \(z-a}(z-b\ 

we have a two-sheeted surface, ?t = 2. There are two branch- points, z = a and z = b; but 
2=00 is not a branch-point ; so that r=2. At each of the branch-points the two values are 
interchanged, so that m 1 = 2, ??i 2 = 2; thus Q = 2. Hence the connectivity =2-4 + 3 = 1, 
that is, the surface is simply connected. 

The surface can be deformed, as in the example in 169, into a sphere. 
Ex. 2. When the equation is 



we have ?t = 2. There are four branch-points, viz., e t , e 2 , e 3 , oc , so that r = 4 ; and at each 
of them the two values of w are interchanged, so that m g = 2 (for 5 = 1,2, 3, 4), and therefore 
Q = 8- 4 = 4. Hence the connectivity is 4- 4 + 3, that is, 3 ; and the value of p is unity. 

Similarly, the surface associated with the equation 



where U(z] is a rational, integral, algebraical function of degree 2i - 1 or of degree 2i, 
is of connectivity 2wi + l ; so that p = m. The equation 

W 2 == ( 1 _ 2 2)( 1 _^2) 

is of class p=\. The case next in importance is that of the algebraical equation leading to 
the hyperelliptic functions, when (/"is either a quintic or a sextic ; and then p = 2. 

Ex. 3. Obtain the connectivity of the Riemann s surface associated with the equation 

w 3 + ^ 3awz = 1 , 
where a is a constant, (i) when a is zero, (ii) when a is different from zero. 



350 RESOLUTION OF A RIEMANN s SURFACE [178. 

Ex. 4. Shew that, if the surface associated with the equation 

f(w,z) = 0, 

have p. boundary-lines instead of one, and if the equation have the same branch-points 
as in the foregoing proposition, the connectivity is Q- 




179. The consideration of irreducible circuits on the surface at once 
reveals the multiple connection of the surface, the numerical measure of 
which has been obtained. In a Riemann s surface, a simple 
closed circuit cannot be deformed over a branch-point. Let 
A be a branch-point, and let AE... be the branch-line 
having a free end at A. Take a curve ...CED... crossing 
the branch-line at E and passing into a sheet different 
from that which contains the portion CE ; and, if possible, 
let a slight deformation of the curve be made so as to transfer the portion 
CE across the branch-point A. In the deformed position, the curve 
...C E D ... does not meet the branch-line; there is, consequently, no 
change of sheet in its course near A and therefore E D ..., which is the 
continuation of ...C E , cannot be regarded as the deformed position of ED. 
The two paths are essentially distinct ; and thus the original path cannot be 
deformed over the branch-point. 

It therefore follows that continuous deformation of a circuit over a 
branch-point on a Riemann s surface is a geometrical impossibility. 

Ex. Trace the variation of the curve CED, as the point E moves up to A and then 
returns along the other side of the branch-line. 

Hence a circuit containing two or more of the branch-points is irreducible ; 
but a circuit containing all the branch-points is equivalent to a circuit that 
contains none of them, and it is therefore reducible. 

If a circuit contain only one branch-point, it can be continuously deformed 
so as to coincide with the point on each sheet and therefore, being deformable 
into a point, it is a reducible circuit. An illustration has already occurred in 
the case of a portion of winding-surface containing a single winding-point 
(p. 348); all circuits drawn on it are reducible. 

It follows from the preceding results that the Riemann s surface associated 
with a multiform function is generally one of multiple connection ; we shall 
find it convenient to know how it can be resolved, by means of cross-cuts, into 
a simply connected surface. The representative surface will be supposed a 
closed surface with a single boundary ; its connectivity, necessarily odd, being 
2/) + l, the number of cross-cuts necessary to resolve the surface into one 
that is simply connected is 2p ; when these cuts have been made, the simply 
connected surface then obtained will have its boundary composed of a single 
closed curve. 



179.] 



BY CROSS-CUTS 



351 




One or two simple examples of resolution of special Riemann s surfaces will be useful 
in leading up to the general explanation ; in the examples it will be shewn how, in 
conformity with 168, the resolving cross-cuts render irreducible circuits impossible. 

Ex. 1. Let *he equation be 

w 1 = A(z-d)(z-b ](z-c}(z-d\ 

where a, b, c, d are four distinct points, all of finite modulus. The surface is two-sheeted ; 
each of the points a, b, c, d is a branch-point where the two values of w interchange ; and 
so the surface, assumed to have a single boundary, is triply connected, the value of p 
being unity. The branch-lines are two, each connecting a pair of branch-points ; let them 
be ab and cd. 

Two cross-cuts are necessary and sufficient to resolve the surface into one that is 
simply connected. We first make a cross-cut, 
beginning at the boundary S, (say it is in the 
upper sheet), continuing in that sheet and re 
turning to J3, so that its course encloses the 
branch-line ab (but not cd) and meets no branch- 
line. It is a cross-cut, and not a loop-cut, for it 
begins and ends in the boundary ; it is evidently 
a cut in the upper sheet alone, and does not 
divide the surface into distinct portions ; and, 
once made, it is to be regarded as boundary for 
the partially cut surface. 

The surface in its present condition is con 
nected : and therefore it is possible to pass from one edge to the other of the cut just 
made. Let P be a point on it ; a curve that passes from one edge to the other is indicated 
by the line PQR in the upper sheet, RS in the lower, and SP in the upper. Along this 
line make a cut, beginning at P and returning to P ; it is a cross-cut, partly in the 
upper sheet and partly in the lower, and it does not divide the surface into distinct 
portions. 

Two cross-cuts in the triply connected surface have now been made ; neither of them, 
as made, divides the surface into distinct portions, and each of them when made reduces 
the connectivity by one unit ; hence the surface is now simply connected. It is easy to 
see that the boundary consists of a single line not intersecting itself; for beginning 
at P, we have the outer edge of PUT, then the inner edge of 2 QltSP, then the inner 
edge of PTB, and then the outer edge of PSRQP, returning to P. 

The required resolution has been effected. 

Before the surface was resolved, a number of irreducible circuits could be drawn ; a 
complete system of irreducible circuits is composed of two, by 168. Such a system may 
be taken in various ways ; let it be composed of a simple curve C lying in the upper sheet 
and containing the points a and b, and a simple curve D, lying partly in the upper 
and partly in the lower sheet and containing the points a and c ; each of these curves 
is irreducible, because it encloses two branch-points. Every other irreducible circuit 
is reconcileable with these two ; the actual reconciliation in particular cases is effected 
most simply when the surface is taken in a spherical form. 

The irreducible circuit C on the unresolved surface is impossible on the resolved 
surface owing to the cross-cut SPQRS ; and the irreducible circuit D on the unresolved 
surface is impossible on the resolved surface owing to the cross-cut PTB. It is easy 
to verify that no irreducible circuit can be drawn on the resolved surface. 



352 



RESOLUTION 



[179. 



In practice, it is conveniently effective to select a complete system of irreducible 
simple circuits and then to make the cross-cuts so that each of them renders one circuit 
of the system impossible on the resolved surface. 

Ex. 2. If the equation be 



= 4 : (z-e 1 )(z-e.t)(z-e 3 ), 

the branch-points are e ls e 2 , e 3 and oo . When the two-sheeted surface is spherical, and the 
branch-lines are taken to be (i) a line joining e lf e. 2 , and (ii) a line joining e 3 to the South 
pole, the discussion of the surface is similar in detail to that in the preceding example. 

Ex. 3. Let the equation be 

t*-4* (!-)(*-*) <X-*)&*-), 

and for simplicity suppose that AC, X, /* are real quantities subject to the inequalities 



The associated surface is two-sheeted and has a boundary assigned to it ; assuming 
that its sheets are planes, we shall take some point in the finite part of the upper sheet, 
not being a branch-point, as the boundary. There are six . branch-points, viz., 0, 1, K, 
X, /x, co at each of which the two values of w interchange ; and so the connectivity of the 
surface is 5 and its class, p, is 2. The branch-lines can be taken as three, this being 
the simplest arrangement ; let them be the lines joining 0, 1 ; K, X ; /*, oo . 

Four cross-cuts are necessary to resolve the surface into one that is simply connected 
and has a single boundary. They may be obtained as follows. 




Fig. 62. 

Beginning at the boundary L, let a cut LHA be made entirely in the upper sheet 
along a line which, when complete, encloses the points and 1 but no other branch-points ; 
let the cut return to L. This is a cross-cut and it does not divide the surface into 
distinct pieces ; hence, after it is made, the connectivity of the modified surface is 4, and 
there are two boundary lines, being the two edges of the cut LHA. 

Beginning at a point A in LHA, make a cut along ABC in the upper sheet until 
it meets the branch-line /zoo, then in the lower sheet along CSD until it meets the 
branch-line 01, and then in the upper sheet from D returning to the initial point A. 
This is a cross-cut and it does not divide the surface into distinct pieces ; hence, after it 
is made, the connectivity of the modified surface is 3, and it is easy to see that there 
is only one boundary edge, similar to the single boundary in Ex. 1 when the surface 
in that example has been completely resolved. 

Make a loop-cut EFG along a line, enclosing the points K and X but no other branch 
points ; and change it into a cross-cut by making a cut from E to some point B of the 
boundary. This cross-cut can be regarded as BEFGE, ending at a point in its own 
earlier course. As it does not divide the surface into distinct pieces, the connectivity is 
reduced to 2 ; and there are two boundary lines. 



179.] BY CROSS-CUTS 353 

Beginning at a point G make another cross-cut GQPRG, as in the figure, enclosing 
the two branch-points X and p, and lying partly in the upper sheet and partly in the lower. 
It does not divide the surface into distinct pieces : the connectivity is reduced to unity 
and there is a single boundary line. 

Four cross-cuts have been made ; and the surface has been resolved into one that is 
simply connected. 

It is easy to verify : 

(i) that neither in the upper sheet, nor in the lower sheet, nor partly in the 
upper sheet and partly in the lower, can an irreducible circuit be drawn in the resolved 
surface ; and 

(ii) that, owing to the cross-cuts, the simplest irreducible circuits in the unresolved 
surface viz. those which enclose 0, 1 ; 1, K ; *, X ; X, /i ; respectively are rendered 
impossible in the resolved surface. 

The equation in the present example, and the Riemann s surface associated with it, 
lead to the theory of hyperelliptic functions*. 

180. The last example suggests a method of resolving any two-sheeted 
surface into a surface that is simply connected. 

The number of its branch-points is necessarily even, say 2p + 2. The 
branch-lines can be made to join these points in pairs, so that there will be 
p + l of them. To determine the connectivity ( 178), we have n = 2 and, 
since two values are interchanged at every branch-point, H = 2p -f 2 ; so 
that the connectivity is 2p + 1. Then 2p cross-cuts are necessary for the 
required resolution of the surface. 

We make cuts round p of the branch-lines, that is, round all of them but 
one ; each cut is made to enclose two branch-points, and each lies entirely in 
the upper sheet. These are cuts corresponding to the cuts LHA and EFG 
in fig. 62 ; and, as there, the cut round the first branch-line begins and ends 
in the boundary, so that it is a cross-cut. All the remaining cuts are loop- 
cuts at present. The system of p cuts we denote by a 1} a 2 , ..., a p . 

We make other p cuts, one passing from the inner edge of each of the p 
cuts a already made to the branch-line which it surrounds, then in the lower 
sheet to the (j) + l)th branch-line, and then in the upper sheet returning to 
the point of the outer edge of the cut a at which it began. This system of 
cuts corresponds to the cuts ADSGBA and GQPRG in fig. 62. Each of them 
can be taken so as to meet no one of the cuts a except the one in which it 
begins and ends ; and they can be taken so as not to meet one another. 
This system of p cuts we denote by b l} b. 2 , ..., b p , where b r is the cut which 
begins and ends in a r . All these cuts are cross-cuts, because they begin and 
end in boundary-lines. 

Lastly, we make other p 1 cuts from a r to 6. r _ 1} for r = 2, 3, . .., p, all in 

* One of the most direct discussions of the theory from this point of view is given by Prym, 
Neue Theorie der ultraelliptischen Functionen, (Berlin, Mayer and Miiller, 2nd ed., 1885). 

F. 23 






354 GENERAL RESOLUTION OF SURFACE [180. 

the upper sheet ; no one of them, except at its initial and its final points, 
meets any of the cuts already made. This system of p - 1 cuts we denote 
by c% , GS, . . . , Cp . 

Because b r ^ is a cross-cut, the cross-cut c r changes a r (hitherto a loop- 
cut) into a cross-cut when c r and a r are combined into a single cut. 

It is evident that no one of these cuts divides the surface into distinct 
pieces; and thus we have a system of 2p cross-cuts resolving the two-sheeted 
surface of connectivity 2p+I into a surface that is simply connected. The 
cross-cuts in order* are 

Oj, &j, C 2 and a a , 6 2 , c 3 and a s , b s , ...,c p and a p , b p . 

181. This resolution of a general two-sheeted surface suggests f Rie- 

mann s general resolution of a surface with any (finite) number of sheets. 

As before, we assume that the surface is closed and has a single boundary 

and that its class is p, so that 2p cross-cuts are necessary for its resolution 

into one that is simply connected. 

Make a cut in the surface such as not to divide it into distinct pieces; 
and let it begin and end in the boundary. It is a cross-cut, say ^ ; it 
changes the number of boundary-lines to 2 and it reduces the connectivity 
of the cut surface to 2p. 

Since the surface is connected, we can pass in the surface along a 
continuous line from one edge of the cut ^ to the opposite edge. Along 
this line make a cut 6j : it is a cross-cut, because it begins and ends in 
the boundary. It passes from one edge of c^ to the other, that is, from one 
boundary-line to another. Hence, as in Prop. II. of 164, it does not divide 
the surface into distinct pieces; it changes the number of boundaries to 1 
and it reduces the connectivity to 1p 1. 

The problem is now the same as at first, except that now only 
2 2 cross-cuts are necessary for the required resolution. We make a 
loop-cut a. 2 , not resolving the surface into distinct pieces, and a cross-cut 
d from a point of a 2 to a point on the boundary at 6j ; then Cj and a, 2> taken 
together, constitute a cross-cut that does not resolve the surface into distinct 
pieces. It therefore reduces the connectivity to 2p 2 and leaves two pieces 
of boundary. 

The surface being connected, we can pass in the surface along a continuous 
line from one edge of a to the opposite edge. Along this line we make a cut 
b. 2 , evidently a cross-cut, passing, like h in the earlier case, from one 
boundary-line to the other. Hence it does not divide the surface into 

* See Neumann, pp. 178 182; Prym, Zur Thcorie der Fwwtionen in einer zweiblattrigen 
Flfahe, (1866). 

+ Riemann, Ges. Werke, pp. 122, 123 ; Neumann, pp. 182185. 



181.] BY CROSS-CUTS 355 

distinct pieces; it changes the number of boundaries to 1 and it reduces 
the connectivity to 2p 3. 

Proceeding in p stages, each of two cross-cuts, we ultimately obtain a 
simply connected surface with a single boundary ; and the general effect on 
the original unresolved surface is to have a system of cross-cuts somewhat of 
the form 




Fig. 63. 

The foregoing resolution is called the canonical resolution of a Riemann s 
surface. 

Ex. 1. Construct the Riemann s surface for the equation 

w 3 + z 3 3awz 1, 

both for a = and for a different from zero; and resolve it by cross-cuts into a simply 
connected surface with a single boundary, shewing a complete system of irreducible simple 
circuits on the unresolved surface. 

Ex. 2. Shew that the Riemann s surface for the equation 



_ 
(z-c)(z-d) 

is of class p = 2- indicate the possible systems of branch-lines, and, for each system, 
resolve the surface by cross-cuts into a simply connected surface with a single boundary. 

(Burnside.) 

182. Among algebraical equations with their associated Riemann s 
surfaces, two general cases of great importance and comparative simplicity 
distinguish themselves.. The first is that in which the surface is two- 
sheeted ; round each branch-point the two branches interchange. The 
second is that in which, while the surface has a finite number of sheets 
greater than two, all the branch-points are of the first order, that is, are 
such that round each of them only two branches of the function interchange. 
The former has already been considered, in so for as concerns the surface ; 
we now proceed to the consideration of the latter. 

The equation is f(w, z) = 0, 

of degree n in w; and, for our present purpose, it is convenient to regard 

as an equation corresponding to a generalised plane curve of degree n 
so that no term in / is of dimensions higher than n. 

The total number of branch-points has been proved, in 98, to be 

w(w-l)-28-2, 

232 



356 DEFICIENCY [182. 

where S is the number of points which are the generalisation of double 
points on the curve with non-coincident tangents and K is the number 
of double points on the curve with coincident tangents. Round each of 
these branch-points, two branches of w interchange and only two, so that 
all the numbers m q of 178 are equal to 2 ; hence the ramification 
H is 

2 [n (n - 1) - 2S - 2/e} - [n (n - 1) - 2S - 2*}, 

that is, n=w(n-l)-28-2. 

The connectivity of the surface is therefore 

w (n - 1) - 28 - 2* - 2n + 3 ; 
and therefore the class p of the surface is 

(n-l)(-2)-8-. 

Now this integer is known* as the deficiency of the curve; and therefore it 
appears that the deficiency of the curve is the same as the class of the Riemann 
surface associated with its equation, and also is the same as the class of its 
equation. 

Moreover, the number of branch-points of the original equation is fl, that 

is, 

n - 2 



Note. The equality of these numbers, representing the deficiency and 
the class, is one among many reasons that lead to the close association of 
algebraic functions (and of functions dependent on them) with the theory of 
plane algebraic curves, in the investigations of Nb ther, Brill, Clebsch and 
others, referred to in 191, 242. 

183. With a view to the construction of a canonical form of Riemann s 
surface of class p for the equation under consideration, it is necessary to 
consider in some detail the relations between the branches of the functions 
as they are affected by the branch- points. 

The effect produced on any value of the function by the description of a 
small circuit, enclosing one branch-point (and only one), is known. But 
when the small circuit is part of a loop, the effect on the value of the 
function with which the loop begins to be described depends upon the form 
of the loop; and various results (e.g. Ex. 1, 104) are obtained by taking 
different loops. In the first form ( 175) in which the branch-lines were 
established as junctions between sheets, what was done was the equivalent 

* Salmon s Higher Plane Curves, 44, 83; Clebsch s Vorlesungen iiber Geometrie, (edited 
by Lindemann), t. i, pp. 351 429, the German word used instead of deficiency being Geschlecht. 
The name deficiency was introduced by Cayley in 1865: see Proc. Land. Math. Soc., vol. i., 
" On the transformation of plane curves." 



183.] LOOPS 357 

of drawing a number of straight loops, which had one extremity common to 
all and the other free, and of assigning the law of junction according to the 
law of interchange determined by the description of the loop. As, however, 
there is no necessary limitation to the forms of branch-lines, we may draw 
them in other forms, always, of course, having branch-points at their free 
extremities ; and according to the variation in the form of the branch-line, 
(that is, according to the variation in the form of the corresponding loop 
or, in other words, according to the deformation of the loop over other 
branch-points from some form of reference), there will be variation in the law 
of junction along the branch-lines. 

There is thus a large amount of arbitrary character in the forms of the 
branch-lines, and consequently in the laws of junction along the branch-lines, 
of the sheets of a Riemann s surface. Moreover, the assignment of the n 
branches of the function to the n sheets is arbitrary. Hence a consider 
able amount of arbitrary variation in the configuration of a Riemann s 
surface is possible within the limits imposed by the invariance of its 
connectivity. The canonical form will be established by making these 
arbitrary elements definite. 

184. After the preceding explanation and always under the hypothesis 
that the branch-points are simple, we shall revert temporarily to the use of 
loops and shall ultimately combine them into branch-lines. 

When, with an ordinary point as origin, we construct a loop round a 
branch-point, two and only two of the values of the function are affected 
by that particular loop ; they are interchanged by it ; but a different form of 
loop, from the same origin round the same branch-point, might affect some 
other pair of values of the function. 

To indicate the law of interchange, a symbol will be convenient. If the 
two values interchanged by a given loop be Wi and w m , the loop will be 
denoted by im ; and i and ra will be called the numbers of the symbol of that 
loop. 

For the initial configuration of the loops, we shall (as in 175) take an 
ordinary point : we shall make loops beginning at 0, forming them in the 
sequence of angular succession of the branch-points round and drawing the 
double linear part of the loop as direct as possible from to its branch-point : 
and, in this configuration, we shall take the law of interchange by a loop to 
be the law of interchange by the branch-point in the loop. 

In any other configuration, the symbol of a loop round any branch-point 
depends upon its form, that is, depends upon the deformation over other 
branch-points which the loop has suffered in passing from its initial form. 
The effect of such deformation must first be obtained : it is determined by 
the following lemma : 



358 



MODIFICATION 



[184. 



When one loop is deformed over another, the symbol of the deformed loop is 
unaltered, if neither of its numbers or if both of its numbers occur in the 
symbol of the unmoved loop ; but if, before deformation, the symbols have one 
number common, the new symbol of the deformed loop is obtained from the old 
symbol by substituting, for the common number, the other number in the symbol 
of the unmoved loop. 

The sufficient test, to which all such changes must be subject, is that 
the effect on the values of the function at any point of a contour enclosing 
both branch-points is the same at that point for all deformations into two 
loops, Moreover, a complete circuit of all the loops is the same as a contour 
enclosing all the branch-points; it therefore (Cor. III. 90) restores the initial 
value with which the circuit began to be described. 

Obviously there are three cases. 

First, when the symbols have no number common : let them be mn, rs. 
The branch-point in the loop rs does not affect w m or w n : it is thus effectively 
not a branch-point for either of the values w m and w n ; and therefore ( 91) 
the loop mn can be deformed across the point, that is, it can be deformed 
across the loop mn. 

Secondly, when the symbols are the same : the symbol of the deformed 
loop must be unaltered, in order that the contour embracing only the two 
branch-points may, as it should, restore after its complete description each of 
the values affected. 

Thirdly, when the symbols have one number common : let be any 
point and let the loops be OA, OB in any given position such as (i), Fig. 64, 
with symbols mr, nr respectively. Then OB may be deformed over OA as 
in (ii), or OA over OB as in (iii). 




Fig. 64 

The effect at of a closed circuit, including the points A and B and 
described positively beginning at 0, is, in (i) which is the initial configura 
tion, to change w m into w r , w r into w n , w n into w m \ this effect on the 
values at 0, unaltered, must govern the deformation of the loops. 

The two alternative deformations (ii) and (iii) will be considered separately. 

When, as in (ii), OB is deformed over OA, then OA is unmoved and 
therefore unaltered : it is still mr. Now, beginning at with w m , the loop 



184.] OF LOOPS 359 

OA changes w m into w r : the whole circuit changes w m into w r , so that OB 
must now leave w r unaltered. Again, beginning with w n , it is unaltered by 
A, and the whole circuit changes w n into w m : hence OB must change w n 
into w m , that is, the symbol of OB must be inn. And, this being so, an 
initial w r at is changed by the whole circuit into w n , as it should be. 
Hence the new symbol mn of the deformed loop OB in (ii) is obtained from 
the old symbol by substituting, for the common number r, the other number 
in in the symbol of the unmoved loop OA. 

We may proceed similarly for the deformation in (iii) ; or the new symbol 
may be obtained as follows. The loop A in (iii) may be deformed to the 
form in (iv) without crossing any branch- point and therefore without 
changing its symbol. When this form of the loop is described in the 
positive direction, w n initially at is changed into w. r after the first loop 
OB, for this loop has the position of OB in (i), then it is changed into w m 
after the loop OA, for this loop has the position of OA in (i), and then w m is 
unchanged after the second (and inner) loop OB. Thus w n is changed into 
w m , so that the symbol is mn, a symbol which is easily proved to give the 
proper results with an initial value w m or w r for the whole contour. This 
change is as stated in the theorem, which is therefore proved. 

Ex. If the deformation from (i) to (ii) be called superior, and that from (i) to (iii) 
inferior, then x successive superior deformations give the same loop-configuration, in 
symbols and relative order for positive description, as 6 & successive inferior deform 
ations. 

COROLLARY. A loop can be passed unchanged over two lo.ops that have the 
same symbol. 

Let the common symbol of the unmoved loops be mn. If neither number 
of the deformed loop be m or n, passage over each of the loops mn makes no 
difference, after the lemma ; likewise, if its symbol be mn. If only one of its 
numbers, say n, be in mn, its symbol is nr, where r is different from m. When 
the loop nr is deformed over the first loop mn, its new symbol is mr ; when 
this loop mr is deformed over the second loop mn, its new symbol is nr, that 
is, the final symbol is the same as the initial symbol, or the loop is unchanged. 

185. The initial configuration of the loops is used by Clebsch and 
Gordan to establish their simple cycles and thence to deduce the periodi 
city of the Abelian integrals connected with the equation f(w, z) = 0, 
without reference to the Riemann s surface ; and this method of treating 
the functions that arise through the equation, always supposed to have 
merely simple branch-points, has been used by Casorati* and Liiroth-J-. 

We can pass from any value of w at the initial point to any other 

* Annali di Matematica, 2 da Ser., t. iii, (1870), pp. 1 27. 
t Abh. d. K. bay. Akad. t. xvi, i Abth., (1887), pp. 199241. 



360 CYCLES OF LOOPS [185. 

value by a suitable series of loops ; because, were it possible to inter 
change the values of only some of the branches, an equation could be 
constructed which had those branches for its roots. The fundamental 
equation could then be resolved into this equation and an equation having 
the rest of the branches for its roots : that is, the fundamental equation 
would cease to be irreducible. 

We begin then with any loop, say one connecting w l with w 2 . There 
will be a loop, connecting the value w 3 with either w l or w.,; there will 
be a loop, connecting the value w t with either w 1} w. 2 , or w 3 ; and so on, 
until we select a loop, connecting the last value w n with one of the other 
values. Such a set of loops, n 1 in number, is called fundamental. 

A passage round the set will not at the end restore the branch with 
which the description began. When we begin with any value, any other 
value can be obtained after the description of properly chosen loops of the 
set. 

Any other loop, when combined with a set of fundamental loops, gives 
a system the description of suitably chosen loops of which restores some 
initial value ; only two values can be restored by the description of loops 
of the combined system. Thus if the loops in order be 12, 13, 14,..., In 
and a loop qr be combined with them, the value w q is changed into Wj_ by 
Iq, into w r by Ir, into w q by qr; and similarly for w r . Such a combination 
of n loops is called a simple cycle. 

The total number of branch-points, a.nd therefore of loops, is ( 182) 

2 {/> + (-!)}; 

and therefore the total number of simple cycles is 2p+n l. But these 
simple cycles are not independent of one another. 

In the description of any cycle, the loops vary in their operation 
according to the initial value of w : and, for two different initial values of 
w, no loop is operative in the same way. For otherwise all the preceding 
and all the succeeding loops would operate in the same way and would 
lead, on reversal, to the same initial value of w. Hence a loop of a given 
cycle can be operative in only two descriptions, once when it changes, say, wi 
into Wj, and the other when it changes Wj into W{. 

Now consider the circuit made up of all the loops. When w l is taken as 
the initial value, it is restored at the end : and in the description only a 
certain number of loops have been operative : the cycle made up of these loops 
can be resolved into the operative parts of simple cycles, that is, into simple 
cycles : hence one relation among the simple cycles is given by the considera 
tion of the operative loops when the whole system of the loops is described 
with an initial value. 

Similarly when any other initial value is taken ; so that apparently there 



185.] LUROTH S THEOREM 361 

are n relations, one arising from each initial value. These n relations are not 
independent : for a simultaneous combination of the operations of all the 
loops in all the circuits leads to an identically null effect (but no smaller 
combination would be effective), for each loop is operative twice (and only 
twice) with opposite effects, shewing that one and only one of the relations is 
derivable from the remainder. Hence there are n 1 independent relations 
and therefore* the number of independent simple cycles is 2p. 

186. We now proceed to obtain a typical form of the Riemann s surface 
by deforming the initial configuration of the loops into a typical configu 
ration f. The final arrangement of the loops is indicated by the two 
theorems : 

I. The loops can be made in pairs in which all loop-symbols are of the 
form (m, in + I), for m = 1, 2, ... , n 1. (With this configuration, w 1 can be 
changed by a loop only into w. 2 , w., by a loop only into w 3 , and so on in 
succession, each change being effected by an even number of loops.) This 
theorem is due to Liiroth. 

II. The loops can be made so that there is only one pair 12, only one 
pair 23, . . . , only one pair ()i 2, n 1 ), and the remaining p + I pairs are 
(n 1, n). This theorem is due to Clebsch. 

187. We proceed to prove Liiroth s theorem, assuming that the loops 
have the initial configuration of 184. 

Take any loop 12, say OA : beginning it with w 1} describe loops positively 
and in succession ; then as the value w l is restored sooner or later, for it 
must be restored by the circuit of all the loops, let it be restored first by a 
loop OB, the symbol of OB necessarily containing the number 1. Between OA 
and OB there may be loops whose symbols contain 1 but which have been 
inoperative. Let each of these in turn be deformed so as to pass back over 
all the loops between its initial position and OA ; and then finally over OA. 
Before passing over OA its symbol must contain 1, for there is no loop over 
which it has passed that, having 1 in its symbol, could make it drop 1 in the 
passage ; but it cannot contain 2, for, if it did, the effect of OA and the 
deformed loop would be to restore 1, an effect that would have been 
caused in the original position, contrary to the hypothesis that OB is the 
first loop that restores 1. Hence after it has passed over OA its symbol 
no longer contains 1. 

* Clebsch und Gordan, Theorie der AbcVschen Functional, p. 85. 
t The investigation is based upon the following memoirs : 

Liiroth, "Note liber Verzweigungsschnitte und Querschnitte in einer Riemann scheu Fla che," 
Math. Ann., t. iv, (1871), pp. 181184; "Ueber die kanonischen Perioden der Abel schen 
Integrate," Abh. d. K. bay. Akad., t. xv, ii Abth., (1885), pp. 329366. 

Clebsch, "Zur Theorie der Riemann schen Flachen," Math. Ann., t. vi, (1873), pp. 216230. 
Clifford, " On the canonical form and dissection of a Riemann s Surface," Loud. Math. Soc. 
Proc., vol. viii, (1877), pp. 292304. 



362 LUROTH S THEOREM [187. 

Next, pass OB over the loops between its initial position and OA but not 
over OA : its symbol must be 12 in the deformed position since w, is restored 
by the loop OB. Then OA and the deformed loop OB are each 12 ; hence each 
of the loops, between the new position and the old position of OB, can be passed 
over OA and the new loop OB without any change in its symbol. There are 
therefore, behind OA, a series of loops that do not affect w^ Thus the loops 



are 



(a) loops behind OA not affecting w lf (b) OA, OB each 12, 
(c) other loops beyond the initial position of OB. 

Begin now with w a at the loop OB and again describe loops positively 
and in succession: then w. 2 must be restored sooner or later. It may be 
only after OA is described, so that there has been a complete circuit of 
all the loops ; or it may first be by an intermediate loop, say 00. 

For the former case, when OA is the first loop by which w. 2 is restored, 
we deform as follows. Deform all loops affecting w 1} which lie between 
OB and OA, in the positive direction from OB back over other loops and 
over OB. The symbol of each just before its deformation contains 1 but 
not 2, and therefore after its deformation it does not contain 1. Moreover 
just after OB is described, w l is the value, and just before OA is described, 
w l is the value ; hence the intermediate loops, which have affected w lf 
must be even in number. Let OG be the first after OB which affects w lt 
and let the symbol of OG be Ir. Then beginning OG with w 1} the value 
Wj_ must be restored by a complete circuit of all the loops, that is, it 
must be restored by OB] and therefore the value must be Wi when 
beginning OA, or Wj. must be restored before OA. Let OH be the first 
loop after OG to restore w^, then, by proceeding as above, we can deform 
all the loops between OG and OH over OG, with the result that no such 
deformed loop affects w and that OG and OH are both Ir. Hence all 
the loops affecting w 1 can be arranged in pairs having the same symbol 

Since OG and OH are a pair with the same symbol, every loop between 
OB and OG can be passed unchanged over OG and OH together. When 
this is done, pass OG over OB so that it becomes 2r, and then OH over 
OB so that it also is 2r. Thus these deformed loops OG, OH are a pair 
2r; and therefore OA can, without change, be deformed over both so as 
to be next to OB. Let this be done with all the pairs ; then, finally, we 
have 

(a) loops not affecting w lt (b) a pair with the symbol 12, 

(c) pairs affecting w, and not w 1} (d) loops not affecting w. 

We thus have a pair 12 and loops not affecting tv^ so that such a change 
has been effected as to make all the loops affecting w 1 possess the symbol 12. 
For the second case, when OC is the first loop to restore w,, the 



187.] ON CONFIGURATION OF LOOPS 363 

value with which the loop OB whose symbol is 12 began to be described, we 
treat the loops between OB and 00 in a manner similar to that adopted in 
the former case for loops between OA and OB ; so that, remembering that 
now w. 2 instead of the former w l is the value dealt with in the recurrence, we 
can deform these loops into 

(a) loops behind OB which change w l but not w. 2 , 

(b) OB and OC, the symbol of each of which is 12. 

Now OB was next to OA ; hence the set (a) are now next to OA. Each of 
them when passed over OA drops the number 1 from its symbol and so the 
whole system now consists of 

(a) loops behind OA not affecting w lt (b) OA, OB, 00 each of which 
is 12, (c) other loops. 

Begin again with the value w l before OA. Before OC the value is w^\ 
and the whole circuit of the loops must restore w 1} which must therefore 
occur before OA. Let OD be the first loop by which w^ is restored. Then 
treating the loops between OC and OD, as formerly those between the initial 
positions of OA and OB were treated, we shall have 

(a) loops behind OA not affecting w n (b) OA, OB each being 12, 

(c) loops between OB and OC not affecting w 1} (d) OC, OD each 

being 12, (e) other loops. 

Except that fewer loops affecting w l have to be reckoned with, the con 
figuration is now in the same condition as at the end of the first stage. 
Proceeding therefore as before, we can arrange that all the loops affecting w t 
occur in pairs with the symbol 12. Moreover, each of the loops in the set 
(c) can be passed unchanged over OA and OB ; so that, finally, we have 

(a) pairs of loops with the symbol 12, (& ) loops not affecting w^. 

We keep (a) in pairs, so that any desired deformation of loops in (& ) over 
them can be made without causing any change; and we treat the set (6 ) in 
the same manner as before, with the result that the set (b ) is replaced by 

(6) pairs of loops with the symbol 23, (c ) loops not affecting W L or w. 2 . 

And so on, with the ultimate result that Hie loops can be made in pairs in 
which each symbol is of the form (in, m + 1) for m = l, ... , n 1. 

188. We now come to Clebsch s Theorem that the loops thus made can 
be so deformed that there is only one pair 12, only one pair 23, and so on, 
until the last symbol (n 1, n), which is the common symbol of p+ 1 pairs. 

This can be easily proved after the establishment of the lemma that, if 
there be two pairs 1 2 and one pair 23, the loops can be deformed into one pair 
12 and two pairs 23. 



364 CLEBSCH S THEOREM [188. 

The actual deformation leading to the lemma is shewn in the accompany 
ing scheme : the deformations implied by the 

i r i c xi i fj. 12 12 12 12 26 2o 

continuous lines are those ot a loop from the left 

to the right of the respective lines, and those 12 12 12 23 13 23 

implied by the dotted lines are those of a loop 12 12 3 13 13 23 
from the right to the left of the respective lines. 

., ,-, . L2 L2 id lo ^o AO 

It is interesting to draw figures, representing 

the loops in the various configurations. 12 23 12 13 23 23 

By the continued use of this lemma we can 12 23 23 12 23 23 

change all but one of the pairs 12 into pairs 23, 12 12 23 23 23 23 
all but one of the pairs 23 into pairs 34, and 

so on, the final configuration being that there are one pair 12, one pair 23, ... 
and p + 1 pairs (n - 1, n). Thus Clebsch s theorem is proved. 

189. We now proceed to the construction of the Biemann s surface. 

Each loop is associated with a branch-point, and the order of interchange 
for passage round the branch-point, by means of the loop, is given by the 
numbers in the symbol of the loop. 

Hence, in the configuration which has been obtained, there are two branch 
points 12: we therefore connect them (as in 176) by a line, not necessarily 
along the direction of the two loops 12 but necessarily such that it can, 
without passing over any branch-point, be deformed into the lines of the 
two loops; and we make this the branch-line between the first and the 
second sheets. There are two branch-points 23 : we connect them by a line 
not meeting the former branch-line, and we make it the branch-line between 
the second and the third sheets. And so on, until we come to the last two 
sheets. There are *2p + 2 branch-points n-l,n: we connect these in pairs 
(as in 176) by p + 1 lines, not meeting one another or any of the former 
lines, and we make them the p + 1 branch-lines between the last two sheets. 

It thus appears that, when the winding -points of a Riemann s surface with 
n sheets of connectivity 2p + 1 are all simple, the surface can be taken in such 
a form that there is a single branch-line between consecutive sheets except for tlie 
last two sheets : and between the last two sheets there are p+l branch-lines. 
This form of Riemann s surface may be regarded as the canonical form for a 
surface, all the branch-points of which are simple. 

Further, let AB be a branch-line such as 12. Let two points P and Q 
be taken in the first sheet on opposite sides of AB, so that PQ in space is 
infinitesimal ; and let P be the point in the second sheet determined by the 
same value of z as P, so that P Q in the sheet is infinitesimal. Then the 
value Wi at P is changed by a loop round A (or round B) into a value at Q 
differing only infinitesimally from w. 2 , which is the value at P : that is, the 
change in the function from Q to P is infinitesimal. Hence the value of the 
function is continuous across a line of passage from one sheet to another. 



190.] CANONICAL SURFACE 365 

190. The class of the foregoing surface is p ; and it was remarked, in 
170, that a convenient surface of reference of the same class is that of a 
solid sphere with p holes bored through it. It is, therefore, proper to in 
dicate the geometrical deformation of a Riemann s surface of this canonical 
form into a p-ho\ed sphere. 

The Riemann s surface consists of n sheets connected chainwise each with 
a single branch-line to the sheet on either side of it, except that the first is 
connected only with the second and that the last two have p + 1 branch- 
lines. We may also consider the whole surface as spherical and the sequence 
of the sheets from the inside outwards : and the outmost sheet can be con 
sidered as bounded. 

Let the branch-line between the first and the second sheets be made to 
lie along part of a great circle. Let the first sheet of the Riemann s surface 
be reflected in the plane of this great circle : the line becomes a long 
narrow hole along the great circle, and the reflected sheet becomes a large 
indentation in the second sheet. Reversing the process of 169, we can 
change the new form of the second sheet, so that it is spherical again : it is 
now the inmost of the n 1 sheets of the surface, the connectivity and the 
ramification of which are unaltered by the operation. 

Let this process be applied to each surviving inner sheet in succession. 
Then, after n-2 operations, there will be left a two-sheeted surface ; the 
outer sheet is bounded and the two sheets are joined by p + 1 branch- 
lines ; so that the connectivity is still 2|) + 1. Let these branch-lines be 
made to lie along a great circle: and let the inner surface be reflected 
in the plane of this circle. Then, after the reflexion, each of the branch-lines 
becomes a long narrow hole along the great circle ; and there are two 
spherical surfaces which pass continuously into one another at these holes, 
the outer of the surfaces being bounded. By stretching one of the holes 
and flattening the two surfaces, the new form is that of a bifacial flat 
surface: each of the p holes then becomes a hole through the body 
bounded by that surface ; the stretched hole gives the extreme geo 
metrical limits of the extension of the surface, and the original boundary of 
the outer surface becomes a boundary hole existing in only one face. The 
body can now be distended until it takes the form of a sphere, and the final 
form is that of the surface of a solid sphere with p holes bored through it 
and having a single boundary. 

This is the normal surface of reference ( 170) of connectivity 2p + 1. 

As a last ground of comparison between the Riemann s surface in its 
canonical form and the surface of the bored sphere, we may consider the 
system of cross-cuts necessary to transform each of them into a simply 
connected surface. 

We begin with the spherical surface. The simplest irreducible circuits 



366 DEFORMATION [190. 

are of two classes, (i) those which go round a hole, (ii) those which go through 
a hole; the cross-cuts, 2p in number, which make the surface simply con 
nected, must be such as to prevent these irreducible circuits. 

Round each of the holes we make a cut a, the first of them beginning 
and ending in the boundary : these cuts prevent circuits through the holes. 
Through each hole we make a cut 6, beginning and ending at a point in the 
corresponding cut a : we then make from the first b a cut Ci to the second a, 
from the second 6 a cut c 2 to the third a, and so on. The surface is then 
simply connected : a is a cross-cut, 6j is a cross-cut, Cj + a 2 is a cross-cut, 
& 2 is a cross-cut, c 2 + a 3 is a cross-cut, and so on. The total number is 
evidently 2p, the number proper for the reduction ; and it is easy to verify 
that there is a single boundary. 

To compare this dissection with the resolution of a Riemann s surface by 
cross-cuts, say of a two-sheeted surface (the rc-sheeted surface was trans 
formed into a two-sheeted surface), it must be borne in mind that only p of 
the p + 1 branch-lines were changed into holes and the remaining one, which, 
after the partial deformation, was a hole of the Riemann s surface, was 
stretched out so as to give the boundary. 

It thus appears that the direction of a cut a round a hole in the normal 
surface of reference is a cut round a branch-line in one sheet, that is, it is a 
cut a as in the resolution ( 180) of the Riemann s surface into one that is 
simply connected. 

Again, a cut b is a cut from a point in the boundary across a cut a and 
through the hole back to the initial point ; hence, in the Riemann s surface, 
it is a cut from some one assigned branch-line across a cut a r , meeting the 
branch-line surrounded by a r , passing into the second sheet and, without 
meeting any other cut or branch-line in that surface, returning to the initial 
point on the assigned branch-line. It is a cut b as in the resolution of the 
Riemann s surface. 

Lastly, a cut c is made from a cut b to a cut a. It is the same as in the 
resolution of the Riemann s surface, and the purpose of each of these cuts is 
to change each of the loop-cuts a (after the first) into cross-cuts. 

A simple illustration arises in the case of a two-sheeted Riemann s surface, of class> = 2. 
The various forms are : 

(i) the surface of a two-holed sphere, with the directions of cross-cuts that resolve it 
into a simply connected surface; as in (i), Fig. 65, B, K being at opposite edges of 
the cut Cj where it meets a. 2 : II, C at opposite edges where it meets b^. and so on; 

(ii) the spherical surface, resolved into a simply connected surface, bent, stretched, 
and flattened out ; as in (ii), Fig. 65; 

(iii) the plane Riemann s surface, resolved by the cross-cuts ; as in Fig. 63, p. 355. 

Numerous illustrations of transformations of Riemann s surfaces are given by 
Hofmann, Methodik der stetigen Deformation von zweibldttrigan Riemann schen Fliichen, 
(Halle a. S., Nebert, 1888). 



190.] 



OF RIEMANNS SURFACES 



367 




191. We have seen that a bifacial surface with a single boundary can be 
deformed, at least geometrically, into any other bifacial surface with a single 
boundary, provided the two surfaces have the same connectivity ; and the 
result is otherwise independent of the constitution of the surface, in regard 
to sheets and to form or position of branch-lines. Further, in all the geo 
metrical deformations adopted, the characteristic property is the uniform 
correspondence of points on the surfaces. 

Now with every Riemann s surface, in its initial form, an algebraical 
equation /(w, z) = Q is associated; but when deformations of the surface 
are made, the relations that establish uniform correspondence between 
different forms, practically by means of conformal representation, are often 
of a transcendental character (Chap. XX.). Hence, when two surfaces are 
thus equivalent to one another, and when points on the surfaces are 
determined solely by the variables in the respective algebraical equations, 
no relations other than algebraical being taken into consideration, the 
uniform correspondence of points can only be secured by assigning a new 
condition that there be uniform transformation between the variables w and 
2 of one surface and the variables w and z of the other surface. And, when 
this condition is satisfied, the equations are such that the deficiencies of the 
two (generalised) curves represented by the equations are the same, because 
they are equal to the common connectivity. It may therefore be expected 
that, when the variables in an equation are subjected to uniform transfor 
mation, the class of the equation is unaltered ; or in other words that the 
deficiency of a curve is an invariant for uniform transformation. 

This inference is correct : the actual proof is more directly connected 
with geometry and the theory of Abelian functions, and must be sought 
elsewhere*. The result is of importance in justifying the adoption of a 
simple normal surface of the same class as a surface of reference. 

* Clebsch s Vorlcsungen iiber Geometric, t. i, p. 459, where other references are given; Salmon s 
Higher Plane Curves, pp. 93, 319; Clebsch und Gordan, Theorie der Abel schen Functionen, 
Section 3; Brill, Math. Ann., t. vi, pp. 33 65. 



CHAPTER XVI. 

ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS. 

192. IN the preceding chapter sufficient indications have been given as 
to the character of the Riemann s surface on which the ?i-branched function 
w, determined by the equation 

/(,*)=(>, 

can be represented as a uniform function of the position of the variable. It 
is unnecessary to consider algebraically multiform functions of position on 
the surface, for such multiformity would merely lead to another surface of 
the same kind, on which the algebraically multiform functions would be 
uniform functions of position ; transcendental ly multiform functions of 
position will arise later, through the integrals of algebraic functions. It 
therefore remains, at the present stage, only to consider the most general 
uniform function of position on the Riemann s Surface. 

On the other hand, it is evident that a Riemann s Surface of any number 
of sheets can be constructed, with arbitrary branch-points and assigned 
sequence of junction ; the elements of the surface being subject merely to 
general laws, which give a necessary relation between the number of sheets, 
the ramification and the connectivity, and which require the restoration of 
any value of the function after the description of some properly chosen 
irreducible circuit. The essential elements of the arbitrary surface, and the 
merely general laws indicated, are independent of any previous knowledge 
of an algebraical equation associated with the surface ; and a question arises 
whether, when a Riemann s surface is given, an associated algebraical equa 
tion necessarily exists. 

Two distinct subjects of investigation, therefore, arise. The first is the 
most general uniform function of position on a surface associated with a given 
algebraical equation, and its integral ; the second is the discussion of the 
existence of functions of position on a surface that is given independently 



192.] FUNCTIONS OF POSITION 369 

of an algebraical equation. Both of them lead, as a matter of fact, to the 
theory of transcendental (that is, non-algebraical) functions of the most 
general type, commonly called Abelian transcendents. But the first is, 
naturally, the more direct, in that the algebraical equation is initially given : 
whereas, in the second, the prime necessity is the establishment of the so- 
called Existence-Theorem that such functions, algebraical and transcen 
dental, exist. 

193. Taking the subjects of investigation in the assigned order, we 
suppose the fundamental equation to be irreducible, and algebraical as 
regards both the dependent and the independent variable ; the general form 
is therefore 

W n G (Z) + W"-^ (Z) + . . . + wG n ^ (Z) + G n (z) = 0, 

the coefficients G (z), G^(z\ ..., G n (z) being rational, integral, algebraical 
functions. 

The infinities of w are, by 95, the zeros of G (z) and, possibly, z= oo . 
But, for our present purpose, no special interest attaches to the infinity of a 
function, as such ; we therefore take wG (z) as a new dependent variable, 
and the equation then is 

/ (w, z) = w n + w n ~ l g 1 (z)+...+ wg n ^ (z) + g n (z) = 0, 

in which the functions g (z) are rational, integral, algebraical functions 
of z. 

The distribution of the branches for a value of z which is an ordinary 
point, and the determination of the branch-points together with the cyclical 
grouping of the branches round a branch-point, may be supposed known. 
When the corresponding w-sheeted Riemann s surface (say of connectivity 
2p + 1) is constructed, then w; is a uniform function of position on the 
surface. 

Now not merely w, but every rational function of w and z, is a uniform 
function of position on the surface; and its branch-points (though not 
necessarily its infinities) are the same as that of the function w. 

Conversely, every uniform function of position on the Riemanris surface, 
having accidental singularities and infinities only of finite order, is an 
algebraical rational function of w and z. The proof* of this proposition, 
to which we now proceed, leads to the canonical expression for the most 
general uniform function of position on the surface, an expression which is 
used in Abel s Theorem in transcendental integrals. 

Let w denote the general uniform function, and let w/, w/, ..., w n denote 
the branches of this function for the points on the n sheets determined by 

* The proof adopted follows Prym, Crelle, t. Ixxxiii, (1877), pp. 251261 ; see also Klein, 
Ueber Riemann s Theorie der algebraischen Functionen und ihrer Integrate, p. 57. 

F. 24 



370 



UNIFORM FUNCTIONS OF POSITION 



[193. 



the algebraical magnitude z\ and let w lt w. 2 , ...,w n be the corresponding 
branches of w for the magnitude z. Then the quantity 

WfWi + W*Wz + + W n *Wn, 

where s is any positive integer, is a symmetric function of the possible values 
of w s w ; it has the same value in whatever sheet z may lie and by whatever 
path z may have attained its position in that sheet ; the said quantity is there 
fore a uniform function of z. Moreover, all its singularities are accidental in 
character, by the initial hypothesis as to w and the known properties of 
w ; they are finite in number ; and therefore the uniform function of z is 
algebraical. Let it be denoted by h, (z), which is an integral function only 
when the singularities are for infinite values of z ; then 

wfWi + wfWz + . . . + w n *w n = h s (z), 

an equation which is valid for any positive integer s, there being of course 
the suitable changes among the rational integral algebraical functions h (z} for 
changes in s. It is unnecessary to take s ^ n, when the equations for the 
values 0,1, ..., n 1 of s are retained: for the equations corresponding to 
values of s ^ n can be derived, from the n equations that are retained, by 
using the fundamental equation determining w. 

Solving the equations 



-iWi + W 2 W 2 -I- . . . + W n W n = h-i 



Wf-hVi + ... + W n "~W = hn-i (z\ 

to determine w/, we have 



1, 


1, .-, 1 


= 


h (z), 1, . 


.., 1 


w,, 


Wo, ..., W n 




/? ^.2^ Wo 


.., w 


w 2 


w.?, ..., w n 2 




*,(*), W 2 2 


w 2 


w n-i y 


w,/*- 1 , , ., Wn"- 1 




^ (^) Wo n ~ 1 


w n1 



The right-hand side is evidently divisible by the product of the differences 
of w 2 , w 3 , ..., w n ; and this product is a factor of the coefficient of w/. 
Then, if 

n 

(w w 2 ) (w w 3 ) ... (w w n ) = S k r w n ~ r , 

r=l 

where ki is unity, we have, on removing the common factor, 

1 (W x - W 2 ) (W 1 W 3 ) . . . (Wj Wn) 



193.] ON A RIEMANN S SURFACE 371 

But / (w, z) = (w w^ (w w 2 )...(w w n ), 

so that k 2 = w l + g l (z), 

k 3 = wf + W& (z) + g 2 (z), 



k n = wf- 1 + w^-g, (z) + . . . + g n ^ (z). 

When these expressions for k are substituted in the numerator of the ex 
pression for Wi, it takes the form of a rational integral algebraical function 
of w of degree n 1 and of z, say 

h (z) w^- 1 -f H, (z) < l ~ 2 + . . . + H n _, (z) w, + H n ^ (z). 

The denominator is evidently df/dw l , when w is replaced by w l after differen 
tiation, so that we now have 






df/dw, 

The corresponding form holds for each of the branches of w : and therefore we 
have 

) w n ~* + . . . + ffn-i (z) 



df/dw 



= 



nw n ~ l + (n - 1) w n ~ z g l (z)+...+ g n _, (z) 

so that w is a rational, algebraical, function of w and z. The proposition is 
therefore proved. 

By eliminating w between f (w, z) = and the equation which expresses 
w in terms of w and z, or by the use of 99, it follows that w satisfies an 
algebraical equation 

F(w ,z) = 0, 

where F is of order n in w ; the equations / (w, z) and F (w , z) = have 
the same Riemann s surface associated with them*. 

194. It thus appears that there are uniform functions of position on 
the Riemann s surface just as there are uniform functions of position in 
a plane. The preceding proposition is limited to the case in which the 
infinities, whether at branch-points or not, are merely accidental ; had the 
function possessed essential singularities, the general argument would still 
be valid, but the forms of the uniform functions h (z) would no longer be 
algebraical. In fact, taking account of the difference in the form of the 
surface on which the independent variable is represented, we can extend 
to multiform functions, which are uniform on a Riemann s surface, those 
propositions for uniform functions which relate to expansion near an ordinary 
point or a singularity or, by using the substitution of 93, a branch 
singularity, those which relate to continuation of functions, and so on ; 

* See 191. Functions related to one another, as w and w are, are called gleichverzweigt, 
Riemann, p. 93. 

242 



372 ALGEBRAICAL FUNCTIONS [194. 

and their validity is not limited, as in Cor. VI., 90, to a portion of the 
surface in which there are no branch-points. 

Thus we have the theorem that a uniform algebraical function of position 
on the Riemanns surface has as many zeros as it has infinities. 

This theorem may be proved as follows. 

The function is a rational algebraical function of w and z. If it be also integral, 
let it be itf=U (w, z), where U is integral. 

Then the number of the zeros of w on the surface is the number of simultaneous roots 
common to the two equations U (w, z) = 0,f(w, z) = Q. If u^ and/ M denote the aggregates 
of the terms of highest dimensions in these equations say of dimensions X and /j. respec 
tively then Xfj. is the number of common roots, that is, the number of zeros of w 1 . 

The number of points, where it/ assumes a value A, is the number of simultaneous 
roots common to the equations U (w, z) = A, f(w, z) = Q, that is, it is X/x as before. Hence 
there are as many points where vf assumes a given value as there are zeros of w \ and 
therefore the number of the infinities is the same as the number of zeros. The number 
of infinities can also be obtained by considering them as simultaneous roots common to 

^=0,^=0. 

If the function be not integral, it can (193) be expressed in the form w = ->-- - .- , where 

V (W, Z) 

U and V are integral, rational algebraical functions. The zeros of uf are the zeros of U 
and the infinities of F, the numbers of which, by what precedes, are respectively the same 
as the infinities of U and the zeros of V. The latter are the infinities of vf; and therefore 
w has as many zeros as it has infinities. 

Note. When the numerator and the denominator of a uniform fractional 
function of z have a common zero, we divide both of them by their greatest 
common measure; and the point is no longer a common zero of their new 
forms. But when the numerator U (w, z) and the denominator V(w, z) of a 
uniform function of position on a Riemann s surface have a common zero, so 
that there are simultaneous values of w and z for which both vanish, U and V 
do not necessarily possess a rational common factor ; and then the common 
zero cannot be removed. 

It is not difficult to shew that this possibility does not affect the preceding theorem. 

195. In the case of uniform functions it was seen that, as soon as theii 
integrals were considered, deviations from uniformity entered. Special inves 
tigations indicated the character of the deviations and the limitations 
their extent. Incidentally, special classes of functions were introduced, 
such as many-valued functions, the values differing by multiples of a 
constant ; and thence, by inversion, simply-periodic functions were deduced. 

So, too, when multiform functions denned by an algebraical equation are 
considered, it is necessary to take into special account the deviations from 
uniformity of value on the Riemann s surface which may be introduced by 
processes of integration. It is, of course, in connexion with the branch 
points that difficulties arise ; but, as the present method of representing the 
variation of the variable is distinct from that adopted in the case of uniform 



195.] PATHS OF INTEGRATION 373 

functions, it is desirable to indicate how we deal with, not merely branch 
points, but also singularities of functions when the integrals of such functions 
are under consideration. In order to render the ideas familiar and to avoid 
prolixity in the explanations relating to general integrals, we shall, after 
one or two propositions, discuss again some of the instances given in 
Chapter IX., taking the opportunity of stating general results as occasion 
may arise. 

One or two propositions already proved must be restated : the difference 
from the earlier forms is solely in the mode of statement, and therefore the 
reasoning which led to their establishment need not be repeated. 

I. The path of integration between any two points on a Riemann s surface 
can, without affecting the value of the integral, be deformed in any possible 
continuous manner that does not make the path pass over any discontinuity of 
the subject of integration. 

This proposition is established in 100. 

II. A simple closed curve on a Riemann s surface, which is a path of 
integration, can, without affecting the value of the integral, be deformed in 
any possible continuous manner that does not make the curve pass over any 
discontinuity of the subject of integration. 

Since the curve on the surface is closed, the initial and the final points 
are the same ; the initial branch of the function is therefore restored after 
the description of the curve. This proposition is established in Corollary II., 
100. 

III. If the path of integration be a curve between two points on different 
sheets, determined by the same algebraical value of z, the curve is not a closed 
curve ; it must be regarded as a path betiueen the two points ; its deformation 
is subject to Proposition I. 

No restatement, from Chapter IX., of the value of an integral, along 
a path which encloses a branch-point, is necessary. The method of dealing 
with the point when that value is infinite will be the same as the method of 
dealing with other infinities of the function. 

196. We have already obtained some instances of multiple-valued 
functions, in the few particular integrals in Chapter IX. ; the differences in 
the values of the functions, arising as integrals, consist solely of multiples of 
constants. The way in which these constants enter in Riemann s method is 
as follows. 

When the surface is simply connected, there is no substantial difference 
from the previous theory for uniform functions ; we therefore proceed to the 
consideration of multiply connected surfaces. 

On a general surface, of any connectivity, take any two points z and z. 
As the surface is one of multiple connection, there will be at least two 



374 CONSTANT OF INTEGRATION [196. 

essentially distinct paths between z and z, that is, paths which cannot be 
reduced to one another ; one of these paths can be deformed so as to be 
made equivalent to a combination of the other with some irreducible circuit. 
Let z 1 denote the extremity of the first path, and let z. 2 denote the same point 
when regarded as the extremity of the second ; then the difference of the 
two paths is an irreducible circuit passing from z to z%. When this circuit 
is made impossible by a cross-cut G passing through the point z, then z l 
and # 2 ma y be regarded as points on the opposite edges of the cross-cut : and 
the irreducible circuit on the unresolved surface becomes a path on the 
partially resolved surface passing from one edge of the cross-cut to the other. 

When the surface is resolved by means of the proper number of cross-cuts 
into a simply connected surface, there is still a path in the surface from 
z-i to 2 on opposite edges of the cross-cut G : and all paths between z l and 
-Z 2 in the resolved surface are reconcileable with one another. One such path 
will be taken as the canonical path from z l to 2 ; it evidently does not meet 
any of the cross-cuts, so that we consider only those paths which do not 
intersect any cross-cut. 

If then Z be the function of position on the surface to be integrated, the 
value of the integral for the first path from z to z^ is 



f 

> Z 



Zdz; 

e*t 

and for the second path it is I Zdz, 

J Z 

or, by the assigned deformation of the second path, it is 

, /"Si 

Zdz + Zdz, 



fZ 

J t 



the second integral being taken along the canonical path from z 1 to z 2 in the 
surface, that is, along the irreducible circuit of canonical form, which would be 
possible in the otherwise resolved surface were the cross-cut G obliterated. 
The difference of the values of the integral is evidently 



/; 



Zdz, 

rz 
which is therefore the change made in the value of the integral I Zdz, 

J Zo 

when the upper limit passes from one edge of the cross-cut to the other ; let 
it be denoted by /. As the curve is, in general, an irreducible circuit, this 
integral / may not, in general, be supposed zero. 

We can arbitrarily assign the positive and the negative edges of some one 
cross-cut, say A. The edges of a cross-cut B that meets A are defined to be 
positive and negative as follows : when a point moves from one edge of B to 
the other, by describing the positive edge of A in a direction that is to the 
right of the negative edge of A, the edge of B on which the point initially 



196.] AT A CROSS-CUT 375 

lies is called its positive edge, and the edge of B on which the point finally 
lies is called its negative edge. And so on with the cross-cuts in succession. 

The lower limit of the integral determining the modulus for a cross-cut 
is taken to lie on the negative edge, and the upper on the positive edge. 

Kegarding a point on the cross-cut as defining two points ^ and z on 
opposite edges which geometrically are coincident, we now prove that for all 
points on the cross-cut which can be reached from without passing over any 
other cross-cut, when the surface is resolved into one that is simply connected, 
the integral I is a constant. For, if be such a point, defining z/ and z 2 on 
opposite edges, then z 1 z. 2 z. 2 2: 1 2 1 is a circuit on the simply connected surface, 
which can be made evanescent ; and it will be assumed that no infinities of Z 
lie in the surface within the circuit, an assumption which will be taken into 
account in 197, 199. Therefore the integral of Z, taken round the circuit, 
is zero. Hence 



Zdz + Zdz + Zdz + Zdz = 0, 

/% J Z 2 Jf, 

that is, p Zdz - \ Zdz = \ Zdz - [" Zdz. 

J Z, J Zi J Zi J 2 2 

Along the direction of the cross-cut, the function Z is uniform : and 
therefore Zdz is the same for each element of the two edges, so long as the 
cross-cut is not met by any other. Hence the sums of the elements on the 
two edges are the same for all points on the cross-cut that can be reached 
from without meeting a new cross-cut. The two integrals on the right- 
hand side of the foregoing equation are equal to one another, and therefore 
also those on the left-hand side, that is, 

f*2 C^ 

Zdz = Zdz, 

J Sl * 2/ 

which shews that the integral I is constant for different points on a portion of 
cross-cut that is not met by any other cross-cut. 

If however the cross-cut be met by another cross-cut C , two cases arise 
according as C" has only one extremity, or has both extremities, on C. 

First, let C have only one extremity on C. By what precedes, the 
integral is constant along OP, and it is constant 
along OQ ; but we cannot infer that it is the same R 

constant for the two parts. The preceding proof 
fails in this case ; the distance z.,z.^ in the resolved 
surface is not infinitesimal, and therefore there is Q 



no element Zdz for z^zj to be the same as the Q z \ O ?j 

element for z^ . Let 7 2 be the constant for OP, 7j Fig. 66. 

that for QO ; and let QP be the negative edge. Then 



,= Zdz, 1,= Zdz. 



376 MODULUS OF PERIODICITY [196. 

Let / be the constant value for the cross-cut OR, and let OR be the 
negative edge ; then 

/z/ 

/ =/ Zdz. 

j ~ 

> i> 2 

In the completely resolved surface, a possible path from z. 2 to z% is z 2 to z 1} z^ 
to Zi, z^ to / ; it therefore is the canonical path, so that 

/-I fZl f2 2 

Zdz + Zdz+\ Zdz 
-1 /% J Z, 

= - / 3 + /! + r zdz. 

JM, 

f* 1 

But I Zdz is an integral of a uniform finite function along an infinitesimal 



arc z-)z{, and it is zero in the limit when we take z y and / as coincident. 
Thus 

/ = !,-/ 

or the constant for the cross-cut OR is the excess of the constant for the part of 
PQ at the positive edge of OR over the constant for the part of PQ at the 
negative edge. 

Secondly, let G have both extremities on G, close to one another so that 
they may be brought together as in the figure : it 
is effectively the case of the directions of two cross 
cuts intersecting one another, say at 0. Let I 1 , I 2 , 
I s , 7 4 be the constants for the portions QO, OP, OR, 
SO of the cross-cuts respectively, and let z 3 z 2 be 
the positive edge of QOP ; then zz 3 is the positive 
edge of SOR. Then if (z) denote the value of 

f z 
the integral I Zdz at 0, which is definite because 

J 2 tl 8 " 

the surface is simply connected and no discontinuities of Z lie within the 
paths of integration, we have 



7 2 = r Zdz = (> 2 ) - <H) (^) ; 

1 
and / 3 = !" Zdz = (*,) - @ (z,), 7 4 = [" Zdz = (z,) - (z,} ; 

J 2 2 J Z, 

so that /! / 2 = / 3 / 4 , 

or the excess of the constant for the portion of a cross-cut on the positive edge, 
over the constant for the portion on the negative edge, of another cross-cut is 
equal to the excess, similarly estimated, for that other cross-cut. 

Ex. Consider the constants for the various portions of the cross-cuts in the canonical 
resolution ( 180, 181) of a Riemann s surface. Let the constants for the two portions 
of a r be A r , A r ; and the constants for the two portions of b r be B r , B r ; and let the 
constant for c r be C r . 



196.] FOR A CROSS-CUT 

Then, at the junction of c r and a r + lt we have 

at the junction of c r and b r , we have 

and, at the crossing of a r and b r , we have A 

Now, because b l is the only cross-cut which meets a lt 
we have A 1 = A 1 ; hence B l = Bj 1 , and therefore C v = 0. 
Hence A 2 = A 2 ; therefore B^=-B^ and therefore also 
<7., = 0. And so on. 



377 




Fig. 68. 



Hence the constant for each of the portions of a cross-cut a is the same ; the constant for 
each of the portions of a cross-cut b is the same ; and the constant for each cross-cut c is zero. 
A single constant may thus be associated with each cross-cut a, and a single constant with 
each cross-cut b, in connexion with the integral of a given uniform function of position on 
the Riemann s surface. It has not been proved and it is not necessarily the fact that 
any one of these constants is different from zero ; but it is sufficiently evident that, if all 
the constants be zero, the integral is a uniform function of position on the surface, that is, 
a rational algebraical function of w and z. 

197. Hence the values of the integral at points on opposite edges of a 
cross-cut differ by a constant. 

Suppose now that the cross-cut is obliterated : the two paths to the point 
z will be the same as in the case just considered and will furnish the same 
values respectively, say U and U + 1. But the irreducible circuit which 
contributes the value / can be described any number of times ; and 
therefore, taking account solely of this irreducible circuit and of the cross-cuts 
which render other circuits impossible on the resolved surface, the general 
value of the integral at the point z is 

U+kl, 
where k is an integer and U is the value for some prescribed path. 

The constant / is called* a modulus of periodicity. 

It is important that every modulus of periodicity should be finite; the path 
which determines the modulus can therefore pass through a point c where 
Z = GO , or be deformed across it without change in the modulus, only if the 
limit of (z c) Z be a uniform zero at the point. If, however, the limit of 
(z c) Z at the point be a constant, implying a logarithmic infinity for the 
integral, or if it be an infinity of finite order (the order not being necessarily 
an integer), implying an algebraical infinity for the integral, we surround 
the point c by a simple small curve and exclude the internal area from the 
range of variation of the independent variable-}-. This exclusion is secured 
by making a small loop-cut in the surface round the point; it increases 
by unity the connectivity of the surface on which the variable is represented. 

* Sometimes the modulus for the cross-cut. 

t This is the reason for the assumption made on p. 375. 



378 THE NUMBER OF INDEPENDENT MODULI [197. 

When the limit of (z c)Z is a uniform zero at c, no such exclusion 
is necessary: the order of the infinity for Z is easily seen to be a proper 
fraction and the point to be a branch-point. 

Similarly, if the limit of zZ for z = co be riot zero and the path which 
determines a modulus can be deformed so as to become infinitely large, it is 
convenient to exclude the part of the surface at infinity from the range of 
variation of the variable, proper account being taken of the exclusion. The 
reason is that the value of the integral for a path entirely at infinity (or 
for a point-path on Neumann s sphere) is not zero ; z = <x> is either a 
logarithmic or an algebraic infinity of the function. But, if the limit of zZ 
be zero for = oo , the exclusion of the part of the surface at infinity is 
unnecessary. 

198. When, then, the region of variation of the variable is properly 
bounded, and the resolution of the surface into one that is simply connected 
has been made, each cross-cut or each portion of cross-cut, that is marked off 
either by the natural boundary or by termination in another cross-cut, 
determines a modulus of periodicity. The various moduli, for a given 
resolution, are therefore equal, in number, to the various portions of the 
cross-cuts. Again, a system of cross-cuts is susceptible of great variation, 
not merely as to the form of individual members of the system (which does 
not affect the value of the modulus), but in their relations to one another. 
The total number of cross-cuts, by which the surface can be resolved into one 
that is simply connected, is a constant for the surface and is independent of 
their configuration : but the number of distinct pieces, defined as above, is 
not independent of the configuration. Now each piece of cross-cut furnishes 
a modulus of periodicity ; a question therefore arises as to the number of 
independent moduli of periodicity. 

Let the connectivity of the surface be N+ 1, due regard being had to the 
exclusions, if any, of individual points in the surface : in order that account 
may be taken of infinite values of the variable, the surface will be assumed 
spherical. The number of cross-cuts necessary to resolve it into a surface 
that is simply connected is N; whatever be the number of portions of the 
cross-cuts, the number of these portions is not less than N. 

When a cross-cut terminates in another, the modulus for the former and 
the moduli for the two portions of the latter are connected by a relation 



= &),- too 



so that the modulus for any portion can be expressed linearly in terms of 
the modulus for the earlier portion and of the modulus for the dividing 
cross-cut. 



198.] IS EQUAL TO THE CONNECTIVITY 379 

Similarly, when the directions of two cross-cuts intersect, the moduli of 
the four portions are connected by a relation 

and by passing along one or other of the cross-cuts, some relation is obtainable 
between w l and &>/ or between w 2 and to./, so that, again, the modulus of any 
portion can be expressed linearly in terms of the modulus for the earlier 
portion and of moduli independent of the intersection. 

Hence it appears that a single constant must be associated with each 
cross-cut as an independent modular constant ; and then the constants 
for the various portions can be linearly expressed in terms of these inde 
pendent constants. There are therefore N linearly independent moduli of 
periodicity: but no system of moduli is unique, and any system can be 
modified partially or wholly, if any number of the moduli of the system be 
replaced by the same number of independent linear combinations of members 
of the system. These results are the analytical equivalent of geometrical 
results, which have already been proved, viz., that the number of independent 
simple irreducible circuits in a complete system is N, that no complete 
system of circuits is unique, and that the circuits can be replaced by 
independent combinations reconcileable with them. 

199. If, then, the moduli of periodicity of a function U at the cross-cuts 
in a resolved surface be I l} / 2 , ..., I N , all the values of the function at 
any point on the unresolved surface are included in the form 

7T_I_ /n-t T _i_ -m T 4. _L T 

where m 1 ,m 2 , ..., in x are integers. 

Some special examples, treated by the present method, will be useful in leading up to 
the consideration of integrals of the most general functions of position on a Riemann s 
surface. 

Ex. 1. Consider the integral I . 

J z 

The subject of integration is uniform, so that the surface is one-sheeted. The origin 
is an accidental singularity and gives a logarithmic infinity for the integral ; it is therefore 
excluded by a small circle round it. Moreover, the value of the integral round a circle 
of infinitely large radius is not zero: and therefore 2 = 00 is excluded from the range of 
variation. The boundary of the single spherical sheet can be taken to be the point 
s= co ; and the bounded sheet is of connectivity 2, owing to the small circle at the origin. 
The surface can be resolved into one that is simply connected by a single cross-cut drawn 
from the boundary at 200 to the circumference of the small circle. 

If a plane surface be used, this cross-cut is, in effect, a section ( 103) of the plane 
made from the origin to the point 2 = 00. 

There is only one modulus of periodicity : its value is evidently / , taken round the 
origin, that is, the modulus is 2ni. Hence whenever the B 

path of variation from a given point to a point z passes ::-.-.-.-....--..-.::.-- ^=Q 

from A to B, the value of the integral increases by 2ni ; but A 

if the path pass from B to A, the value of the integral ^o- fi9 - 

decreases by 2ni. Thus A is the negative edge, and B the positive edge of the cross-cut. 



380 EXAMPLES [199. 

If, then, any one value of I - be denoted by w, all values at the point in the 

J Zo z 
unresolved surface are of the form w + 2mni, where in is an integer ; when z is regarded 

as a function of w, it is a simply- periodic function, having 2n-t for its period. 

Ex. 2. Consider I -^ ^. The subject of integration is uniform, so that the surface 

consists of a single sheet. There are two infinities a, each of the first order, because 
(z + a)Z is finite at these two points : they must be excluded by small circles. The limit, 
when s= co , of z/(z z - a 2 ) is zero, so that the point z = co does not need to be excluded. We 
can thus regard one of the small circles as the boundary of the surface, which is then 
doubly connected : a single cross-cut from the other circle to the boundary, that is, in 
effect, a cross-cut joining the two points a and - a, resolves the surface into one that is 
simply connected. 

It is easy to see that the modulus of periodicity is : that A is the negative edge and 

B the positive edge of the cross-cut : and that, if w be 

a value of the integral in the unresolved surface at any _ tt B - fl 

point, all the values at that point are included in the A 

form Fig. 70. 



where n is an integer. 



Ex. 3. Consider J (a 2 - z 2 )~* dz. The subject of integration is two- valued, so that the 
surface is two-sheeted. The branch-points are +a, and oo is not a branch-point, so that 
the single branch-line between the sheets may be taken as the straight line joining a 
and -a. The infinities are a; but as (z + a) (a 2 -2 2 )~ i vanishes at the points, they do 

not need to be excluded. As the limit of z (a 2 -2 2 )~ i , for 2 = 00, is not zero, we exclude 
z=oo by small curves in each of the sheets. 

Taking the surface in the spherical form, we assign as the boundary the small curve 
round the point 2 = 00 in one of the sheets. The connectivity of the surface, through its 
dependence on branch-lines and branch-points, is unity : owing to the exclusion of the point 
2= co by the small curve in the other sheet, the connectivity is increased by one unit: the 
surface is therefore doubly connected. A single cross-cut will resolve the surface into one 
that is simply connected : and this cross-cut must pass from the boundary at 2 = 00 which 
is in one sheet to the excluded point 2=00 . 

Since the (single) modulus of periodicity is the value of the integral along a circuit in 

the resolved surface from one edge of the cross-cut to _____ .. 

the other, this circuit can be taken so that in the un- R ,-/- " ""^ 



resolved surface it includes the two branch-points ; C-^~~ 

and then, by II. of 195, the circuit can be deformed 

until it is practically a double straight line in the upper 

sheet on either side of the branch line, together with two 

small circles round a and - a respectively. Let P be the 

origin, practically the middle point of these straight lines. O 

Consider the branch (a 2 - 2 2 ) ~ * belonging to the upper Fig . 71- 

sheet. Its integral from P to a is 



*32 



fa 

/ 

Jo 



(a 2 - 2 2 ) 



From a to -a the branch is -(a 2 -* 2 )"* ; the point R is contiguous in the surface, 



199.] OF MODULI OF PERIODICITY 381 

not to P, but (as in 189) to the point in the second sheet beneath P at which the branch is 

(a 2 -2 2 ) , the other branch having been adopted for the upper sheet. Hence, from a 
to a by R, the integral is 

/_2 2\ fJ~ 
( Or Z I U4. 



From - a to Q, the branch is + (a 2 2 2 ) 2 , the same branch as at P : hence from - a to Q, 
the integral is 

f ( 2 -2 2 )~ i ^. 
J -a 

The integral, along the small arcs round a and round a respectively, vanishes for each. 
Hence the modulus of periodicity is 

f "(a 2 - z*)~*dz + f~ a - (a 2 - 2 2 ) -i dz + f (a 2 - z^ dz, 

Jo Jo, J -a 

that is, it is 2rr. 

This value can be obtained otherwise thus. The modulus is the same for all points 
on the cross-cut; hence its value, taken at where 2 = 00, is 

J(a 2 -2 2 )~ i ^, 

passing from one edge of the cross-cut at (J to the other, that is, round a curve in the 
plane everywhere at infinity. This gives 

2ni Lt 2( 2 -2 2 )~* = ^ = 27r, 

3=OO 

the same value as before. 

The latter curve round (7, from edge to edge, can easily be deformed into the former 
curve round a and - a from edge to edge of the cross-cut. 

Again, let w 1 be a value of the integral for a point z 1 in one sheet and ? 2 be a value for 
a point 2 a in the other sheet with the same algebraical value as z 1 : take zero as the 
common lower limit of the integral, being the same zero 
for the two integrals. This zero may be taken in either 
sheet, let it be in that in which z l lies : and then 




Pig . 72 . 

To pass from to z 2 for w 2 , any path can be justifiably deformed into the following: 
(i) a path round either branch-point, say a, so as to return to the point under in the 
second sheet, say to 2 , (ii) any number m of irreducible circuits round a and -a, always 
returning to 2 in the second sheet, (iii) a path from 2 to 2 2 lying exactly under the path 
from to 2 X for w t . The parts contributed by these paths respectively to the integral w 2 
are seen to be 

fa, ro 

(i) a quantity + TT, arising from J Q (a 2 - 2 2 ) ~^ dz + j ^ - (a 2 - 2 2 )~* dz, for reasons 

similar to those above ; 
(ii) a quantity i27r, where m is an integer positive or negative ; 



(iii) a quantity I " - (a 2 - 2 2 )~* dz. 

J 02 



In the last quantity the minus sign is prefixed, because the subject of integration is 
everywhere in the second sheet. Now z 2 = z lt and therefore the quantity in (iii) is 

- [* 



that is, it is - w v ; hence w 2 = (2m + 1 ) TT - 



382 MODULI OF PERIODICITY [199. 

If then we take ?0= / (a 2 z 2 )~^ dz, the integral extending along some denned curve from 

J o 

an assigned origin, say along a straight line, the values of w belonging to the same 
algebraical value of z are 2nn + w or (2m + l)r w; and the inversion of the functional 

relation gives 

(j) (w) =z = (f> {(2mr + w) 




where m and n are any integers. 

Ex. 4. Consider I - r , assuming |c|>|a . The surface is two-sheeted, 
j ( z -o)(a*-#} 

with branch-points at a but not at QC : hence the line joining a and -a is the sole 

branch-line. The infinities of the subject of integration are a, a, and c.. Of these a 

and -a need not be excluded, for the same reason that 

their exclusion was not required in the last example. But 

c must be excluded ; and it must be excluded in both 

sheets, because z = c makes the subject of integration 

infinite in both sheets. There are thus two points of 

accidental singularity of the subject of integration ; in 

the vicinity of these points, the two branches of the 

subject of integration are 

_!_(_<*)-*+..., _ J_ (a ._ cS) -i_... ) FJ g- 73 

Z C 6 C 

the relation between the coefficients of (z - c)" 1 in them being a special case of a more 
general proposition ( 210). And since z/{(z - c) (a 2 - z 2 )*} when 2=00 is zero, oo does not 
need to be excluded. 

The surface taken plane is doubly connected, as in the last example, one of the curves 
surrounding c, say that in the upper sheet, being taken as the boundary of the surface. 
A single cross-cut will suffice to make it simply connected : the direction of the cross-cut 
must pass from the c-curve in the lower sheet to the branch-line and thence to the 
boundary in the upper sheet. 

There is only a single modulus of periodicity, being the constant for the single cross-cut. 
This modulus can be obtained by means of the curve AB in the first sheet; and, on 
contraction of the curve (by II, 195) so as to be infmitesimally near c, it is easily seen to be 
2-Trt (a 2 - c 2 )~*, or say 2n- (c 2 - 2 )~^. But the modulus can be obtained also by means of 
the curve CD; and when the curve is contracted, as in the previous example, so as 
practically to be a loop round a and a loop round -a, the value of the integral is 

dz 



[ a 

J- 



which is easily proved to be 2n-(c 2 a 2 )~ a . 

As in Ex. 4, a curve in the upper sheet which encloses the branch-points and the 
branch-lines can be deformed into the curve AB. 

Ex.b. Consider w=$(4z 3 -g<,z-g 3 y~* dz=$udz. 

The subject of integration is two-valued, and therefore the Riemann s surface is two- 
sheeted. The branch-points are z = co , e lt e%, e 3 where e lt e%, e 3 are the roots of 

4s 3 -#2Z-# 3 = 0; 
and no one of them needs to be excluded from the range of variation of the variable. 



199.] 



OF INTEGRALS 



383 



The connectivity of the surface is 3, so that two cross-cuts are necessary to resolve 
the surface into one that is simply connected. The configurations of the branch-lines and 
Q 2 




Fig. 74. 

of the cross-cuts admit of some variety; two illustrations of branch-lines are given in 
Fig. 74, and a point on Q : in each diagram is taken as boundary. 

The modulus for the cross-cut Q 1 say from the inside to the outside can be obtained 
in two different ways. First, from P, a point on Q l , draw a line to e 2 in the first sheet, 
then across the branch-line, then in the second sheet to e s and across the branch-line, 
then in the first sheet round e 3 and back to P : the circuit is represented by the double 
line between e 2 and e z . The value of the integral is 



f e a C&2 f e a 

I udz+ I (-u)dz, that is, 2 I udz. 
J e t J e a J <? 2 



Again, it can be obtained by a line from P , another point on Q lt to oo , round the branch 
point there and across the branch-line, then in the second sheet to e l and round e lt then 
across the branch-line and back to P : the value of the 
integral is 

P 
JL-2 j udz. 

J i 
But the modulus is the same for P as for P : hence 



f fe s 

= 2 udz = 2 udz. 

J e l J e 2 




This relation can be expressed in a different form. The 
path from e 2 to e 3 can be stretched into another form 
towards 2 = 00 in the first sheet, and similarly for the 
path in the second sheet, without affecting the value of 
the integral. Moreover as the integral is zero for 2 = 00, 
we can, without affecting the value, add the small part 
necessary to complete the circuits from e 2 to oo and from e 3 to oo . 
circuits being given by the arrows, we have 




Fig. 75. 
The directions of these 



or, if 

for X = l, 2, 3, we have* 

say 

and E l is the modulus of periodicity for the cross-cut Q l . 

* See Ex. 6, 104. 



fe, r fe, 

I udz = 2 / udz + 2 I udz, 
J e* J <? 2 J oo 

/ 
ttcfe, 

ps 

E l = 2j^udz = E 2 -E s , 



384 EXAMPLES [199. 

In the same way the modulus of periodicity for Q 2 is found to be 

r fe, 

E 3 =2 I udz and to be 2 I udz, 
J e, J <? 2 

the equivalence of which can be established as before. 

Hence it appears that, if to be the value of the integral at any point in the surface, 



the general value is of the form w + mE 1 + nE 3 , where m and n are integers. As the 
integral is zero at infinity (and for other reasons which have already appeared), it is 
convenient to take the fixed limit z so as to define w by the relation 

w= I udz. 



Now corresponding to a given algebraical value of z, there are two points in the surface 
and two values of w : it is important to know the relation to one another of these two 
values. Let z denote the value in the lower sheet : then the path from z to oo can be 
made up of 

(i) a path from z 1 to oo ; (ii) any number of irreducible circuits from oo to oo ; and 
(iii) across the branch-line and round its extremity to oo . 

These parts respectively contribute to the integral 

.-QO roc 

(i) a quantity I (-u)dz, that is, -I udz, or, -w; (ii) a quantity mE l + nE 3 , 

J z J z 

where m and n are integers ; (iii) a quantity zero, since the integral vanishes 

at infinity : so that w = mE 1 + nE 3 -w. 

If now we regard z as a function of w, say z = $> (M>), we have 



But z z algebraically, so that we have 

= $> (w) = jp (mE 1 + 7iE 3 u- 
as the function expressing z in terms of w. 
Similarly it can be proved that 



the upper and the lower signs being taken together. Now g> (w), by itself, determines a 
value of z, that is, it determines two points on the surface : and $ (w) has different values 
for these two points. Hence a point on the surface is uniquely determined by fp(w) an 



Ex. 6. Consider w = j {(1 -z 2 ) (I - t?s?)}-* dz = J udz. The subject of integration is 

two-valued, so that the surface is two-sheeted. The branch-points are 1, j but 
not oo ; no one of the branch-points need be excluded, nor need infinity. 

The connectivity is 3, so that two cross-cuts will render the surface simply connected : 
let the branch-lines and the cross-cuts be taken as in the figure. 

The details of the argument follow the same course as in the previous case. 

The modulus of periodicity for Q 2 is 2 I udz = 4 I udz = 4K, in the ordinary 



notation. 



i 
The modulus of periodicity for Q l is 2 / udz = 2iA", as before. 



199.] 



OF MODULI OF PERIODICITY 



385 




Hence, if w be a value of the integral for a point z in the first sheet, a more general 
value for that point is w+m4K + n2iK . 

Let uf be a value of the integral for a point 2 in the second sheet, where z" is 
algebraically equal to z the point in the 
first sheet at which the value of the integral 
is w then 

w = 2K + m4K -f n2iK w, 
so that, if we invert the functional relation 
and take z = snw, we have Fig. 76. 

sn w = z = sn (w + 4mK+ 2niK ) 

= sn {(4m + 2) K+ 2niK - w}. 

Ex. 7. Consider the integral w = I- Z - , where u = f(I-z*)(l- 

J (zc)u 

As in the last case, the surface is two-sheeted : the branch-points are 1, + - 1 but no 
one of them need be excluded, nor need z = ao . But the point z=c must be excluded in 
both sheets ; for expanding the subject of integration for points in the first sheet in the 
vicinity of z = c, we have 

1 _, 

z c 
and for points in the second sheet in the vicinity of z=c, we have 

1 -i 

Z 

in each case giving rise to a logarithmic infinity for z = c. 

We take the small curves excluding z = c in both sheets as the boundaries of the 
surface. Then, by Ex. 4 178, (or because one of these curves may be regarded as a 




Fig. 77. 

; boundary of the surface in the last example, and the curve excluding the infinity in the 
other .sheet is the equivalent of a loop-cut which ( 161) increases the connectivity by 
unity), the connectivity is 4. The cross-cuts necessary to make the surface simply 
connected are three. They may be taken as in the figure ; ft is drawn from the boundary 
in one sheet to a branch-line and thence round * to the boundary in the other sheet: Q, 
beginning and ending at a point in ft, and ft beginning and ending at a point in ft. 
The moduli of periodicity are : 

for ft, the quantity (Q 1 = ) 2 W{(1- C 8)(1-^ C 2)}-1, obtained by taking a small curve 
round c in the upper sheet : 



F. 



ft, the quantity (fl^^^-, obtained by taking a circuit round 1 



and p passing from one edge of ft to the other at F: 



25 



386 MODULI OF PERIODICITY [199. 

Q s , the quantity (O 3 = )2 f k . _* , obtained by taking a circuit round -1 

and -T, passing from one edge of <? 3 to the other at G: 
Ic 

so that, if any value of the integral at a point be w, the general value at the point is 



where m n m 2 , m 3 are integers. 

Conversely, z is a triply-periodic function of w; but the function of w is not uniform 
( 108). 

Ex. 8. As a last illustration for the present, consider 



The surface is two-sheeted ; its connectivity is 3, the branch-points being 1, ^ but not 
z = oo . No one of the branch-points need be excluded, for the integral is finite round each 
of them. To consider the integral at infinity, we substitute z=^ , and then 

dz 



giving for the function at infinity an accidental singularity of the first order in each 
sheet. 

The point 2=00 must therefore be excluded from each sheet: but the form of w, for 
infinitely large values of z, shews that the modulus for the cross-cut, which passes from 
one of the points (regarded as a boundary) to the other, is zero. 

The figure in Ex. 6 can be used to determine the remaining moduli. The modulus 
for $ is 



fi /1-&V, 

I-? r dx 
J-l\ I-* 2 



I 



=4 r ^=^nfe 



with the notation of Jacobian elliptic functions. The modulus for Q t is 

i 

= 2 I ( -f] dx 

Jl\ L-X J 



on transforming by the relation % 2 + V = l : the last expression can at once be changed 
into the form 2z (K - -")> witn the same notation as before. 

If then w be any value of the integral at a point on the surface, the general value 

there is 

w + 4mE+ 2m (K - E \ 

where m and n are integers. 



200.] INTEGRAL OF ALGEBRAIC FUNCTION 387 

200. After these illustrations in connection with simple cases, we may 
proceed with the consideration of the integral of the most general function 
w of position on a general Riemann surface, constructed in connection with 
the algebraical equation 

/ (w, z) = w n + w n ~ l g, (*) + ... + wg n ^ (z) + g n (z) = 0, 

where the functions g(z) are rational, integral and algebraical. Subsidiary 
explanations, which are merely generalised from those inserted in the 
preceding particular discussions, will now be taken for granted. 

Taking w in the form of 193, we have 

m > = \ *. (,) + " ( *> """ + df + h -< (*> = 1 h, w + ffi^ , 

dw dw 

1 f 
so that in taking the integral of w we shall have a term - I h (z) dz, where 

n j 

h (z) is a rational algebraical function. This kind of integral has been 
discussed in Chapter II.; as it has no essential importance for the present 
investigation, it will be omitted, so that, without loss of generality merely 
for the present purpose*, we may assume h (z)to vanish; and then the 
numerator of w is of degree not higher than n 2 in w. 

The value of z is insufficient to specify a point on the surface : the values 
of w and z must be given for this purpose, a requisite that was unnecessary 
in the preceding examples because the point z was spoken of as being in the 
upper or the lower of the two sheets of the various surfaces. Corresponding 
to a value a of z, there will be n points : they may be taken in the form 
(a 1} ocj), (a 2 , a 2 ), ..., (a n , a n ), where a l , ..., a n are each algebraically equal to a, 
and ttj, ..., a n are the appropriately arranged roots of the equation 

f(w, a) = 0. 
The function w to be integrated is of the form , where U is of 

/ 

dw 

degree n 2 in w, but though algebraical and rational it is not necessarily 
integral in z. 

An ordinary point of w , which is neither an infinity nor a branch-point, 
is evidently an ordinary point of the integral. 

The infinities of the subject of integration are of prime importance. 
They are: 

(i) the infinities of the numerator, 
(ii) the zeros of the denominator. 
The former are constituted by (a), the poles of the coefficients of powers of w 

* See 207, where h (z) is retained. 

252 



388 INTEGRALS [200. 

in U (w, z), and (ft), z = cc: this value is included, because the only infinities 
of w, as determined by the fundamental equation, arise for infinite values of 
z, and infinite values of w and of z may make the numerator U(w, z) 
infinite. 

So far as concerns the infinities of w which arise when z=<x> (and there 
fore w = oo ), it is not proposed to investigate the general conditions that the 
integral should vanish there. The test is of course that the limit, for z = oo , 

of - y z should vanish for each of the n values of w. 

"L 

dw 

But the establishment of the general conditions is hardly worth the 
labour involved ; it can easily be made in special cases, and it will be 
rendered unnecessary for the general case by subsequent investigations. 

201. The simplest of the instances, less special than the examples 
already discussed, are two. 

The first, which is really that of most frequent occurrence and is of very 
great functional importance, is that in which / (w, z) = has the form 

where S (z) is of order 2m 1 or 2m and all its roots are simple : then 

7)f zU (w z} 

J-= 2w = ^S(z}. In order that the limit of jar* 1 ma y be zero when 

dw J_ 

dw 

z = oo , we see (bearing in mind that U, in the present case, is independent of 
w) that the excess of the degree of the numerator of U over its denominator 
may not be greater than m - 2. In particular, if U be an integral function 
of z, a form of U which would leave fw dz zero at z = oo is 
U = c z m - 2 + dz m ~ 3 + . . . + c m - 3 z + c m _ 2 . 

As regards the other infinities of Uj^8(z\ they are merely the roots of 
S(z) = Q or they are the branch-points, each of the first order, of the 
equation 

By the results of 101, the integral vanishes round each of these points ; and 
each of the points is a branch-point of the integral function. The integral is 
finite everywhere on the surface: and the total number of such integrals, 
essentially different from one another, is the number of arbitrary coefficients 
in U, that is, it is ml, the same as the class of the Riemann s surface 
associated with the equation. 

202. The other important instance is that in which the fundamental 
equation is, so to speak, a generalised equation of a plane curve, so that g s (z) 
is an integral algebraical function of z of degree s : then it is easy to see that, 



202.] OF ALGEBRAIC FUNCTIONS 389 

at z=oc , each branch w^z, so that ^*"^*: hence U (w, z) can vary only 
as 2 n ~ s , in order that the condition may be satisfied. If then U(w, z) be an 
integral function of z, it is evident that it can at most take a form which 
makes U= the generalised equation of a curve of degree n - 3; while, if it be 
V (w, z) 
z J c > then V ( w > z \ supposed integral in z, can at most take a form which 

makes V= the generalised equation of a curve of degree n 2. 

Other forms are easily obtainable for accidental singularities of coefficients 
of w in U (w, z) that are of other orders. 

As regards the other possible infinities of the integral, let c be an acci 
dental singularity of a coefficient of some power of w in U(w, z) ; it may be 

7}f 
assumed not to be a zero of j*- . Denote the n points on the surface by 

(CL kj), (c 2 , & 2 ), ..., (c n , k n ), where d, c 2 , ..., c n are algebraically equal to c. 
In the vicinity of each of these points let w be expanded: then, near (c r ,k r } 
we have a set of terms of the type 



, 



(Z - 

where P(z-c r ) is a converging series of positive integral powers of z-c r . 
A corresponding expansion exists for every one of the n points. 

The integral of w will therefore have a logarithmic infinity at (c r , k r ), 
unless A lif is zero; and it will have an algebraical infinity, unless all the 
coefficients A^ r , ...... , A m>r are zero. 

The simplest cases are 

(i) that in which the integral has a logarithmic infinity but no 

algebraical infinity ; and 
(ii) that in which the integral has no logarithmic infinity. 

For the former, w is of the form ^f?* and therefore in the vicinity of c r 

(g-c)- 

dw 

we have w = *- + P (z - c r ), 



z-c r 



W (k c} 
the value of A 1>r being y ". an d W is an integral function of k r , of 

dh 
degree not higher than n - 2. Hence 



r=l r =l y_ 

dk r 



dk r 



390 INFINITIES OF THE INTEGRAL [202. 

since c is the common algebraical value of the quantities c 1; c 2 , . .., c n . Now 
&J, & 2 , ...,k n are the roots of 



an equation of degree n, while W is of degree not higher than n 2 ; hence, 
by a known theorem*, 

- W(k r ,c) 

r^~~ 

dk r 

n 

so that S A lt r = Q. 

r=l 

The validity of the result is not affected if some of the coefficients A vanish. 
But it is evident that a single coefficient A cannot be the only non-vanishing 
coefficient , and that, if all but two vanish, those two are equal and opposite. 

This result applies to all those accidental singularities of coefficients of 
powers of w in the numerator of w which, being of the first order, give rise 
solely to logarithmic infinities in the integral of w . It is of great importance 
in regard to moduli of periodicity of the integral. 

(ii) The other simple case is that in which each of the coefficients 
A l>r vanishes, so that the integral of w has only an algebraical infinity at 
the point c r , which is then an accidental singularity of order less by unity 
than its order for w . 

In particular, if in the vicinity of c r , the form of w be 



the integral has an accidental singularity of the first order. 
It is easy to prove that 

n 

Zt A 2> r = 0, 
r=l 

so that a single coefficient A cannot be the only non-vanishing coefficient ; 
but the result is of less importance than in the preceding case, for all the 
moduli of periodicity of the integral at the cross-cuts for these points vanish. 
And it must be remembered that in order to obtain the subject of integration 
in this form, some terms have been removed in 200, the integral of which 
would give rise to infinities for either finite or infinite values of . 

It may happen that all the coefficients of powers of w in the numerator 
of w are integral functions of . Then 2 = oo is their only accidental 
singularity ; this value has already been taken into account. 

* Burnside and Panton, Theory of Equations, (3rd ed.), p. 319. 



203.] OF AN ALGEBRAIC FUNCTION 391 

203. The remaining source of infinities of w , as giving rise to possible 
infinities of the integral, is constituted by the aggregate of the zeros of 

I/ 1 

^- = 0. Such points are the simultaneous roots of the equations 



In addition to the assumption already made that /= is the equation of a 
generalised curve of the nth order, we shall make the further assumptions 
that all the singular points on it are simple, that is, such that there are only 
two tangents at the point, either distinct or coincident, and that all the 
branch-points are simple. 

The results of 98 may now be used. The total number of the points 
given as simultaneous roots is n (n 1) : the form of the integral in the 
immediate vicinity of each of the points must be investigated. 

Let (c, 7) be one of these points on the Riemann s surface, and let 
(c + , 7 + v) be any point in its immediate vicinity. 

Q / \ 

I. If ~ - do not vanish at the point, then (c, 7) is a branch-point 
for the function w. We then have 

f (w, z) = A % + B \r + quantities of higher dimensions, 

for points in the vicinity of (c, 7), so that uf when | is sufficiently small. 

Then 

7\f 

r- = %B v + quantities of higher dimensions 

*{, 

when | is sufficiently small. Hence, for such values, the subject of integra 
tion is a constant multiple of 

U (% c ) + positive integral powers of v and 

* + powers of with index > \ 

that is, of "*, when || is sufficiently small. The integral is therefore a 
constant multiple of f * when | is sufficiently small; and its value is there 
fore zero round the point, which is a branch-point for the function repre 
sented by the integral. 

^/* / \ 

II. If ^ J vanish at the point, we have (with the assumptions 
of 98), 

/ (w, z) = A* + 2B& 4 6V + terms of the third and higher degrees ; 
and there are two cases. 

(i) If B* < AC, the point is not a branch-point, and we have 
Cv + Bt; = (B- -ACY- + integral powers 2 , 3 , . . . 



392 INFINITIES OF ALGEBRAIC FUNCTION [203. 

as the relation between v and deduced from/= 0. Then 

7)f 

^- = 2 (B + Cv) + terms of second and higher degrees 

= X" + higher powers of . 
In the vicinity of (c, 7), the subject of integration is 

U (7, c) + Dv + E% + positive integral powers 
\% + higher powers of 

Hence when it is integrated, the first term is ^ log , and the remain- 

A. 

ing terms are positive integral powers of : that is, such a point is a 
logarithmic infinity for the integral, unless U (7, c) vanish. 

If, then, we seek integrals which have not the point for a logarithmic 
infinity and we begin with U as the most general function possible, we can 
prevent the point from being a logarithmic infinity by choosing among the 
arbitrary constants in U a relation such that 



There are S such points ( 98); and therefore 8 relations among the 
constants in the coefficients of U must be chosen, in order to prevent the 
integral 



I 

J 



dw 

from having a logarithmic infinity at these points, which are then ordinary 
points of the integral. 

(ii) If IP = AC, the point is a branch-point ; we have 

+ Cv = ^L^ + M^ + N^ + ... 
as the relation between and v deduced from / = 0. In that case, 

rlf 

^- = 2 (B + Cv) + terms of the second and higher degrees 

a 

= Til* + powers of having indices > f . 
In the vicinity of (c, 7), the subject of integration is 

^ (7> c ) + D V + Et, + higher powers 

a ~ 

L% + higher powers of f 
Hence when it is integrated, the first term is 2 ~ - ~*, and it can be 

_L/ 

proved that there is no logarithmic term ; the point is an infinity for the 
integral, unless U (y, c) vanish. 

If, however, among the arbitrary constants in U we choose a relation such 
that 

U (% c) = 0, 



203.] TO BE INTEGRATED 393 

then the numerator of the subject of integration 

= Dv + E% + higher positive powers 
X " -f /u/- + higher powers of , 

on substituting from the relation between v and derived from the funda 
mental equation. The subject of integration then is 

/*+... 



that is, *L+ 

L$ 

the integral of which is 

A/ 
2 y * + positive powers. 

The integral therefore vanishes at the point : and the point is a branch-point 
for the integral. It therefore follows that we can prevent the point from 
being an infinity for the function by choosing among the arbitrary constants 
in U a relation such that 



There are K such points ( 98): and therefore K relations among the 
constants in the coefficients of U must be chosen in order to prevent the 
integral from becoming infinite at these points. Each of the points is a 
branch-point of the integral. 

204. All the possible sources of infinite values of the subject of integra- 

U(w, z) 
tion w , = -p. , have now been considered. A summary of the preceding 

dw 
results leads to the following conclusions relative to fw dz : 

(i) an ordinary point of w is an ordinary point of the integral : 

(ii) for infinite values of z, the integral vanishes if we assign proper 
limitations to the form of U (w, z) : 

(Hi} accidental singularities of the coefficients of powers of w in 
U(w, z) are infinities, either algebraical or logarithmic or both 
algebraical and logarithmic, of the integral : 

(iv) if the coefficients of powers of w in U(w, z) have no accidental 
singularities except for z = <x>, then the integral is finite for 
infinite values of z (and of w) when U (w, z) is the most general 
rational integral algebraical function of w and z of degree n - 3 ; 
but, if the coefficients of powers of w in U (w, z) have an 
accidental singularity of order p, then the integral will be finite 



394 INTEGRALS [204. 

for infinite values of z (and of w) when U(w, z) is the most 
general rational integral algebraical function of w and z, the 
degree in w being not greater than n 2 and the dimensions 
in w and z combined being not greater than n + p 3 : 

(v) those points, at which df/dw vanishes and which are not branch 
points of the function, can be made ordinary points of the 
integral, if we assign proper relations among the constants 
occurring in U (w, z} : 

(vi) those points, at which df/dw vanishes and which are branch 
points of the function, can, if necessary, be made to furnish 
zero values of the integral by assigning limitations to the 
form of U (w, z) ; each such point is a branch-point of the 
integral in any case. 

These conclusions enable us to select the simplest and most important 
classes of integrals of uniform functions of position on a Riemann s surface. 

205. The first class consists of those integrals which do not acquire* 
an infinite value at any point ; they are called integrals of the first kind^. 

The integrals, considered in the preceding investigations, can give rise to 
integrals of the first kind, if the numerator U (w, z) of the subject of integra 
tion satisfy various conditions. The function U(w, z} must be an integral 
function of dimensions not higher than n 3 in w and z, in order that the 
integral may be finite for infinite values of z and for all finite values of z 
not specially connected with the equation / (w, z} = 0; for certain points 
specially connected with the fundamental equation, being 8 + K in number, 
the value of U (w, z) must vanish, so that there must be B + K relations 
among its coefficients. But when these conditions are satisfied, then the 
integral function is everywhere finite, it being remembered that certain 
limitations on the nature of f (w, z) = have been made. 

Usually these conditions do not determine U (w, z) uniquely save as to a 
constant factor ; and therefore in the most general integral of the first kind a 
number of independent arbitrary constants will occur, left undetermined by 
the conditions to which U is subjected. Each of these constants multiplies an 
integral which, everywhere finite, is different from the other integrals so 
multiplied ; and therefore the number of different integrals of the first kind 
is equal to the number of arbitrary independent constants, left undetermined 
in U. It is evident that any linear combination of these integrals, with 

* They will be seen to be multiform functions even on the multiply connected Eiemann s 
surface, and they do not therefore give rise to any violation of the theorem of 40. 

+ The German title is erster Gattung ; and similarly for the integrals of the second kind and 
the third kind. 



205.] OF THE FIRST KIND 395 

constant coefficients, is also an integral of the first kind ; and therefore a 
certain amount of modification of form among the integrals, after they have 
been obtained, is possible. 

The number of these integrals, linearly independent of one another, is 
easily found. Because U is an integral algebraical function of w and z of 
dimensions n 3, it contains -|(n 1) (n 2) terms in its most general form ; 
but its coefficients satisfy 8 + K relations, and these are all the relations that 
they need satisfy. Hence the number of undetermined and independent 
constants which it contains is 



which, by 182, is the class p of the Riemaiin s surface ; and therefore, for the 
present case, the number of integrals, which are finite everywhere on the surface 
and are linearly independent of one another, is equal to the class of the 
Riemanns surface. 

Moreover, the integral of the first kind has the same branch-points as the 
function vu. Though the integral is finite everywhere on the surface, yet its 
derivative w is not so : the infinities of w are the branch- points. 

The result has been obtained on the original suppositions of 98, which 
were, that all the singular points of the generalised curve f(w, z) = are 
simple, that is, only two tangents (distinct or coincident) to the curve can 
be drawn at each such point, and that all the branch-points are simple. 
Other special cases could be similarly investigated. But it is superfluous to 
carry out the investigation for a series of cases, because the result just 
obtained, and the result of 201, are merely particular instances of a general 
theorem which will be proved in Chapter XVIII., viz., that, associated with 
a Riemanns surface of connectivity 2^ + 1, there are p linearly independent 
integrals of the first kind which are finite everywhere on the surface. 

206. The functions, which thus arise out of the integral of an algebraical 
function and are finite everywhere, are not uniform functions of position on 
the unresolved surface. If the surface be resolved by 2p cross-cuts into one 
that is simply connected, then the function is finite, continuous and uniform 
everywhere in that resolved surface, which is limited by the cross-cuts as a 
single boundary. But at any point on a cross-cut, the integral, at the two 
points on opposite edges, has values that differ by any integral multiple of 
the modulus of the function for that cross-cut (and possibly also by integral 
multiples of the moduli of the function for the other cross-cuts). 

Let the cross-cuts be taken as in 181 ; and for an integral of the first 
kind, say W , let the moduli of periodicity for the cross-cuts be 

&>!, o) 2 , ..., a> p , for a 1( a 2 , . 
and w p+l , a) p+2 ,..., w. 2p , for b 1} 



396 INTEGRALS [206. 

respectively ; the moduli for the portions of cross-cuts c 2 , c s , ..., c p have been 
proved to be zero. 

Some of these moduli may vanish ; but it will be proved later ( 231) that 
all the moduli for the cross-cuts a, or all the moduli for the cross-cuts b, cannot 
vanish unless the integral is a mere constant. In the general case, with which 
we are concerned, we may assume that they do not vanish ; and so it follows 
that, if W be a value of an integral of the first kind at any point on the 
Riemanns surface, all its values at that point are of the form 

Zp 

W+ 2 m r a) r , 

r=l 

where the coefficients in are integers. 

The foregoing functions, arising through integrals that are finite every 
where on the surface, will be found the most important from the point of 
view of Abelian transcendents : but other classes arise, having infinities on 
the surface, and it is important to indicate their general nature before passing 
to the proof of the Existence-Theorem. 

207. First, consider an integral which has algebraical, but not logarithmic, 
infinities. Taking the subject of integration, as in the preceding case, to be 
the most general possible, so that arbitrary coefficients enter, we can, by 
assigning suitable relations among these coefficients, prevent any of the 

7)f 
points, given as zeros of ~- = 0, from being infinities of the integral. It 

follows that then the only infinities of the integral will be the points that are 
accidental singularities of coefficients of powers of w in the numerator of the 
general expression for w . These singularities must each be of the second 
order at least : and, in the expansion of w in the vicinity of each of them, 
there must be no term of index 1, the index that leads, on integration, to a 
logarithm. 

Such integrals are called integrals of the second kind. 

The simplest integral of the second kind has an infinity for only a single 
point on the surface, and the infinity is of the first order only : the integral 
is then called an elementary integral of the second kind. After what has 
been proved in 202 (ii), it is evident that an elementary integral of the 
second kind cannot occur in connection with the equation f(w, z) = 0, unless 
the term h (z) of 200 be retained in the expression for w . 

Ex. 1. Adopting the subject of integration obtained in 200, we have 

, 1 ; , , (7(w, z) 

~n o(&) 87~ 

dw 

where U is of the character considered in the preceding sections, viz., it is of degree n - 2 
in w ; various forms of w lead to various forms of h (z) and of U (w, z). 



207.] OF THE SECOND KIND 397 

If -h (z)=.- , and if c be not a singularity of the coefficient of any power of w 

iv (^2 0) 

in U, it is then evident that 

U(w, 



mo 

and the integral on the right-hand side can by choice among the constants be made an 
integral of the first kind. The integral is not, however, an elementary integral of the 
second kind, because z c is an infinity in each sheet. 

Ex. 2. A special integral of the second kind occurs, when we take an accidental 
singularity, say z = c, of the coefficient of some power of w in U(w, z) and we neglect h (z); 
so that, in effect, the subject of integration w is limited to the form 




U being of degree not higher than n - 2 in 10. To the value z = c, there correspond n points 
in the various sheets ; if, in the immediate vicinity of any one of the points, vf be of the 
form 



in that vicinity the integral is of the form 



z-c r 

For such an integral the sum of the coefficients A r is zero : the simplest case arises 
when all but two, say A l and A 2 , of these vanish. The integral is then of the form 

A 

in the vicinity of c t , and of the form 

-A 



^ + P. 2 (z-c 2 ) 
in the vicinity of c 2 . But the integral is not an elementary integral of the second kind. 

208. To find the general value of an integral of the second kind, 
all the algebraically infinite points would be excluded from the Riemann s 
surface by small curves : and the surface would be resolved into one that is 
simply connected. The cross-cuts necessary for this purpose would consist of 
the set of 2p cross-cuts, necessary to resolve the surface as for an integral of 
the first kind, and of the k additional cross-cuts in relation with the curves 
excluding the algebraically infinite points. 

Let the moduli for the former cross-cuts be 

ej, 6 2 , ..., e p , for the cuts a lt a 2 , ..., a p , 

P+I> *p+i, , e 2 p for the cuts b 1} b.,, ..., b p , respectively: 

the moduli for the cuts c are zero. It is evident from the form of the 

integral in the vicinity of any infinite point that, as the integral has only an 



398 ELEMENTARY INTEGRAL [208. 

algebraical infinity, the modulus for each of the k cross-cuts, obtained by a 
curve from one edge to the other round the point, is zero. Hence if one 
value of the integral of the second kind at a point on the surface be E (z), 
all its values at that point are included in the form 

Zp 

E(z)+ 2 n r e r , 

r=l 

where n t , n 2 , ..., n 2p are integers. 

The importance of the elementary integral of the second kind, inde 
pendently of its simplicity, is that it is determined by its infinity, save as to an 
additive integral of the first kind. 

Let EI (z) and E 2 (z) be two elementary integrals of the second kind, 
having their single infinity common, and let a be the value of z at this point ; 
then in its vicinity we have 

^ <*) - ,~i + I&- ) ^ (*) - ~: a + p > (* - a > 

and therefore A 1 E 2 (z)- AJE^(z) is finite at z = a. This new function is 
therefore finite over the whole Riemann s surface : hence it is an integral of 
the first kind, the moduli of periodicity of which depend upon those of E l (z) 
and E, (z}. 

Ex. It may similarly be proved that for the special case in Ex. 2, 207, when the 
integral of the second kind has two simple infinities for the same algebraical value of z in 
different sheets, the integral is determinate save as to an additive integral of the first kind. 

Let a^ and 2 be the two points for the algebraical value a of z ; and let F(z) and G (z) 
be two integrals of the second kind above indicated having simple infinities at a x and a 2 
and nowhere else. 

Then in the vicinity of a : we have 

F(z] = + P 1 (z - a,\ G (z) = 





s 0*1 

so that BF(z) -AG(z) is finite in the vicinity of a^ 
Again, in the vicinity of 2 , we have, by 202, 



so that BF(z)-AG(z) is finite in the vicinity of 2 also. Hence BF(z)- AG(z) is finite 
over the whole surface, and it is therefore an integral of the first kind ; which proves the 
statement. 

It therefore appears that, if F (z) be any such integral, every other integral of the same 
nature at those points is of the form F(z)+ W, where W is an integral of the first kind. 
Now there are p linearly independent integrals of the first kind : it therefore follows that 
there are p + 1 linearly independent integrals of the second kind, having simple infinities 
with equal and opposite residues at two points, (and at only two points), determined by 
one algebraical value of z. 



208.] OF THE SECOND KIND 399 

From the property that an elementary integral of the second kind is 
determined by its infinity save as to an additive integral of the first kind, we 
infer that there are p + l linearly independent elementary integrals of the 
second kind with the same single infinity on the Riemanns surface. 

This result can be established in connection with f(w, z) = as follows. The subject 
of integration is 



\& i*r- ^ 

aw 

where for simplicity it is assumed that a is neither a branch-point of the function 
nor a singular point of the curve f(w, z) = 0, and in the present case U is of degree 
n-l in w. To ensure that the integral vanishes for 3 = 00, the dimensions of U(w,z) 
may not be greater than n-l. Hence U(w, z), in its most general form, is an integral, 
rational, algebraical function of w and z of degree n-l; the total number of terms is 
therefore ( + !), which is also the total number of arbitrary constants. 

In order that the integral may not be infinite at each of the S + K singularities of the 
curve /(w, z) = 0, a relation U(y, c) = must be satisfied at each of them; hence, on this 
score, there are S -f AC relations among the arbitrary constants. 

Let the points on the surface given by the algebraical value a of z be (<%, ni ), ( 2 , a 2 ), 
..., (a n , a n ). The integral is to be infinite at only one of them ; so that we must have 

U(a r , a r ) = 0, 

for r=2, 3, ...,n; and n-l is the greatest number of such points for which U can vanish, 
unless it vanish for all, and then there would be no algebraical infinity. Hence, on this 
score, there are n-l relations among the arbitrary constants in U. 

In the vicinity of z = a, w = a, let 



then we have Q = v ?-+(?+... 

da da 

where ^ is the value of and | that of |", for z- a and 10 = 0. For sufficiently small 
values of || and |f |, we may take 

** 

da * da 
t or such points we have 

U(w t z)=U(a,a)+ v +c+... 
oa oa 



and 



f 

Then unless ^_ LC/L?) = 1 



df (a, a) "" " 
da 



_ = 

U (a, a) 3 (a, a) 8/ 8 (a, a) 

8a 



for (,), and 

3 (a, a) 



400 INTEGRALS [208. 

for ( 2 , a 2 ), ( 3 , 3 ), ..., (, a n ), there will be terms in - in the expansion of the subject of 

integration in the vicinity of the respective points, and consequently there will be 
logarithmic infinities in the integral. Such infinities are to be excluded ; and therefore 
their coefficients, being the residues, must vanish, so that, on this score, there appear to 
be n relations among the arbitrary constants in U. But, as in 210, the sum of the 
residues for any point is zero : and therefore, when n - 1 of them vanish, the remaining 
residue also vanishes. Hence, from this caxxse, there are only n 1 relations among the 
arbitrary constants in U. 

The tale of independent arbitrary constants in U (w, z), remaining after all the 
conditions are satisfied, is 

4w(n+l)-(8 + K )-(w-l) -(n-l) 



as each constant determines an integral, the inference is that there are p + l linearly 
independent elementary integrals of the second kind with a common infinity. 

209. Next, consider integrals which have logarithmic infinities, inde 
pendently of or as well as algebraical infinities. They are called integrals of 
the third kind. As in the case of integrals of the first kind and the second 
kind, we take the subject of integration to be as general as possible so that it 
contains arbitrary coefficients ; and we assign suitable relations among the 
coefficients to prevent any of the points, given as zeros of dfjdw, from becoming 
infinities of the integral. It follows that the only infinities of the integral 
are accidental singularities of coefficients of powers of w in the numerator 
of the general expression for w ; and that, when w is expanded for points in 
the immediate vicinity of such an expression, the term with index 1 must 
occur. 

To find the general value of an integral of the third kind, we should 
first exclude from the Kiemaim s surface all the infinite points, say 

Li, 1/2, ... , Ifj., 

by small curves ; the surface would then be resolved into one that is simply 
connected. The cross-cuts necessary for this purpose would consist of the 
set of 2p cross-cuts, necessary to resolve the surface for an integral of the 
first kind, and of the additional cross-cuts, /JL in number and drawn from the 
boundary (taken at some ordinary point of the integral) to the small curves 
that surround the infinities of the function. 

The moduli for the former set may be denoted by 

CTJ , CTO , . . . , vr p for the cuts a ly a 2 , . . . , a p , 
and vr p+1 , -n-p+2, ..., vT 2p for the cuts b 1} b, ..., b p respectively; 

they are zero for the cuts c. Taking the integral from one edge to the other 
of any one of the remaining cross-cuts l lt L, ..., l q , (where l q is the cross-cut 
drawn from the curve surrounding l q to the boundary), its value is given by 



209.] OF THE THIRD KIND 401 

the value of the integral round the small curve and therefore it is 2-TriX, 
where the expansion of the subject of integration in the immediate vicinity 



of z = l q is 



Then, if II be any value of the integral of the third kind at a point on the 
unresolved Riemann s surface, all its values at the point are included in the 
form 



^ 

11+ 2 m/nv + ZTTI n q \ q , 

r=l 7=1 

where the coefficients m 1 , ..., m 2p , 1} ..., W M are integers. 

210. It can be proved that the quantities \ q are subject to the relation 



Let the surface be resolved by the complete system of 2p + //, cross-cuts : the 
resolved surface is simply connected and has only a single boundary. The 
subject of integration, w , is uniform and continuous over this resolved surface: 
it has no infinities in the surface, for its infinities have been excluded ; hence 

fw dz = 0, 

when the integral is taken round the complete boundary of the resolved 
surface. 

This boundary consists of the double edges of the cross-cuts a, b, c, L, 
and the small curves round the //, points I ; the two edges of the same cross 
cut being described in opposite directions in every instance. 

Since the integral is zero and the function is finite everywhere along the 
boundary, the parts contributed by the portions of the boundary may be con 
sidered separately. 

First, for any cross-cut, say a q : let be the point where it is crossed by b q , 
and let the positive direction of description of the whole boundary be indicated 
by the arrows (fig. 81, p. 438). Then, for the portion Ca...E, the part of the 

rE 

integral is I w dz, or, if Ca. . .E be the negative edge (as in 196), the part of 

w C 

the integral may be denoted by 

/w dz. 
c 

The part of the integral for the portion F...aD, being the positive 

rD (-F 

edge of the cross-cut, is I w dz, which may be denoted by - 1 w dz. The 

J F J D 

course and the range for the latter part are the same as those for the 
F - 26 



402 ELEMENTARY INTEGRAL [210. 

former, and w is the same on the two edges of the cross-cut ; hence the 

sum of the two is 

a 
= \ (w f w ) dz, 

J c 

which evidently vanishes*. Hence the part contributed to fw dz by the two 
edges of the cross-cut a q is zero. 

Similarly for each of the other cross-cuts a, and for each of the cross-cuts 
b, c, L. 

The part contributed to the integral taken along the small curve enclosing 
l q is 2Tri\ q> for q = 1, 2, . . . , /A : hence the sum of the parts contributed to the 
integral by all these small curves is 



All the other parts vanish, and the integral itself vanishes ; hence 

M 

1 ^Q = 0, 



3=1 

establishing the result enunciated. 

COROLLARY. An integral of the third kind, that is, having logarithmic 
infinities on a Riemanns surface, must have at least two logarithmic infinities. 

If it had only one logarithmic infinity, the result just proved would 
require that \ should vanish, and the infinity would then be purely 
algebraical. 

211. The simplest instance is that in which there are only two 
logarithmic infinities ; their constants are connected by the equation 

A.J + X 2 = 0. 

If, in addition, the infinities be purely logarithmic, so that there are no 
algebraically infinite terms in the expansion of the integral in the vicinity 
of either of the points, the integral is then called an elementary integral 
of the third kind. If two points C^ and C 2 on the surface be the two infini 
ties, and if they be denoted by assigning the values Cj and c 2 to z ; and if 
Xj = 1 = X 2 (as may be assumed, for the assumption only implies division 
of the integral by a constant factor), the expansion of the subject of inte 
gration for points in the vicinity of Ci is 

1 



z Cj 

* It vanishes from two independent causes, first through the factor w -w , and secondly 
because z =z n , the breadth of any cross-cut being infinitesimal. 

E C 

The same result holds for each of the cross-cuts a and 6. 

For each of the cross-cuts c and L, the sum of the parts contributed by opposite edges vanishes 
only on account of the factor w - w ; in these cases the variable z is not the same for the upper 
and lower limit of the integral. 



211.] OF THE THIRD KIND 403 

and for points in the vicinity of c a the expansion is 

-1 



z-c 



P 2 (z-c 2 ). 



Such an integral may be denoted by I1 12 : its modulus, consequent on 
the logarithmic infinity, is 



Ex. 1. Prove that, if n 12 , n 23 , H 31 be three elementary integrals of the third kind 
having c lt c. 2 ; c 2 , c 3 ; c 3 , c t for their respective pairs of points of logarithmic discontinuity, 
then n, 2 + n23 + n 31 is either an integral of the first kind or a constant. 

Clebsch and Gordan pass from this result to a limit in which the points c t and c 2 
coincide and obtain an expression for an elementary integral of the second kind in the 
form of the derivative of H 13 with regard to c t . Klein, following Riemann, passes from an 
elementary integral of the second kind to an elementary integral of the third kind by 
integrating the former with regard to its parametric point*. 

Ex. 2. Reverting again to the integrals connected with the algebraical equation 
/(-, z)=Q, when it can be interpreted as the equation of a generalised curve, an integral of 
the third kind arises when the subject of integration is 




where V(w, z) is of degree n- 2 in w. If V(w, z) be of degree in z not higher than n- 2, 
the integral of w is not infinite for infinite values of z; so that V(w, z) is a general integral 
algebraical function of w of degree n 2. 

Corresponding to the algebraical value c of z, there are n points on the surface, say 
( c n ^i)> ( C 2> ^2)? > ( c ni #n); and the expansion of w in the vicinity of (c ri &,.) is 



the coefficients of the infinite terms being subject to the relation 

V(k r ,c r ) 



because V(w, z) is only of degree n - 2 in w. The integral of w will have a logarithmic 
infinity at each point, unless the corresponding coefficient vanish. 

Not more than n- 2 of these coefficients can be made to vanish, unless they all vanish; 
and then the integral has no logarithmic infinity. Let n - 2 relations, say 

V(k r , c r ) = 

for r = 2, 3, ..., n, be chosen ; and let the S + K relations be satisfied which secure that the 
integral is finite at the singularities of the curve / (w, z-) = 0. Then the integral is an 
elementary integral of the third kind, having (c n ^) and (c 2 , 2 ) for its points of 
logarithmic discontinuity. 

Ex. 3. Prove that there are p + l linearly independent elementary integrals of the 
third kind, having the same logarithmic infinities on the surface. 

* Clebsch und Gordan, (I.e., p. 361, note), pp. 2833 ; Klein-Fricke, Vorlesungen iiber die 
Theorie der elliptischen Modulfunctionen, t. i, pp. 518522; Biemann, p. 100. 

262 



404 CLASSES OF FUNCTIONS [211. 

Ex. 4. Shew that, in connection with the fundamental equation 



any integral of the first kind is a constant multiple of 

[dz 

Jw 2 

that an integral of the second kind, of the class considered in Ex. 2, 207, is given by 

\w -, 



and that an elementary integral of the third kind is given by 

^ dz. 



Ex. 5. An elementary (Jacobian) elliptic integral of the third kind occurs in Ex. 7, 
p. 385 ; and a (Jacobian) elliptic integral of the second kind occurs in Ex. 8, p. 386. 

Shew that an elementary (elliptic) integral of the second kind, associated with the 

equation 

v*=4e*-gf-ff d> 

and having its infinity at (c 1? yj, is 

7i ( w 



f 
J 



and that an elementary (elliptic) integral of the third kind, associated with the same 
equation and having its two infinities at (c 1; yj), (c 2 , y 2 )> ^ s 



A sufficient number of particular examples, and also of examples with 
a limited generality, have been adduced to indicate some of the properties 
of functions arising, in the first instance, as integrals of multiform functions 
of a variable z (or as integrals of uniform functions of position on a 
Biemann s surface). The succeeding investigation establishes, from the most 
general point of view, the existence of such functions on a Riemann s 
surface : they will no longer be regarded as defined by integrals of multi 
form functions. 



CHAPTER XVII. 

SCHWARZ S PROOF OF THE EXISTENCE-THEOREM. 

212. THE investigations in the preceding chapter were based on 
the supposition that a fundamental equation was given, the appropriate 
Riemann s surface being associated with it. The general expression of 
uniform functions of position on the surface was constructed, and the 
integrals of such functions were considered. These integrals in general 
were multiform on the surface, the deviation from uniformity consisting 
in the property that the difference between any two of the infinite number of 
values could be expressed as a linear combination of integral multiples of 
certain constants associated with the function. Infinities of the functions 
defined by the integrals, and the classification of the functions according to 
their infinities, were also considered. 

But all these investigations were made either in connection with 
very particular forms of the fundamental equation, or with a form of not 
unlimited generality : and, for the latter case, assumptions were made, 
justified by the analysis so far as it was carried, but not established generally. 

In order to render the consideration of the propositions complete, it must 
be inade without any limitations upon the general form of fundamental 
equation. 

Moreover, the second question of 192, viz., the existence of functions 
(both uniform and multiform) of position on a surface given independently of 
any algebraical equation, is as yet unconsidered. 

The two questions, in their generality, can be treated together. In the 
former case, with the fundamental equation there is associated a Riemann s 
surface, the branching of which is determined by that fundamental equation ; 
in the latter case, the Riemann s surface with assigned branching is supposed 



406 INITIAL SIMPLIFICATION [212. 

given*. We shall take the surface as having one boundary and being other 
wise closed ; the connectivity is therefore an uneven integer, and it will be 
denoted by 2p + 1. 

213. The problem can be limited initially, so as to prevent unnecessary 
complications. All the functions to be discussed, whether they be algebraical 
functions or integrals of algebraical functions, can be expressed in the form 
u + iv, where u and v are two real functions of two independent real variables 
x and y. It has already ( 10) been proved that both u and v satisfy the 
equation 



and that, if either u or v be known, the other can be derived by a quadra 
ture at most, and is determinate save as to an additive arbitrary constant. 
Since therefore w is determined by u, save as to an additive constant, we 
shall, in the first place, consider the properties of the real function u only. 

The result is valid so long as v can be determined, that is, so long as the 
function u has differential coefficients. It will appear, in the course of the 
present chapter, that no conditions are attached to the derivatives of u along 
the boundary of an area, so that the determination of v along such a boundary 
seems open to question. 

It has been ( 36) proved, in a theorem due to Schwarz, that, if w a 
function of z be defined for a half-plane and if it have real finite continuous 
values along any portion of the axis of x, it can be symmetrically continued 
across that portion of the axis. The continuation is therefore possible for the 
real part u of the function w ; and the values of u are the real finite continuous 
values of w along that portion of the axis. 

It will be seen, in Chapters XIX., XX. that, by changing the independent 
variables, the axis of x can be changed into a circle or other analytical line 
(| 221) ; so that a function u, defined for an interior and having real finite con 
tinuous values along any portion of the boundary, can be continued across that 
portion of the boundary, which is therefore not the limit of existence "f of u. 

* The surface is supposed given ; we are not concerned with the quite distinct question as to 
how far a Kiemann s surface is determinate by the assignment of its number of sheets, its 
branch-points (and consequently of its connectivity), and of its branch-lines. This question is 
discussed by Hurwitz, Math. Ann., t. xxxix, (1891), pp. 1 61. He shews that, if ft denote the 
ramification ( 179) of the surface which, necessarily an even integer, is denned as the sum of 
the orders of its branch-points, a two- sheeted surface is made uniquely determinate by assigned 
branch-points; the number of essentially distinct three-sheeted surfaces, made determinate by 
assigned branch-points, is ^(3 n ~ 2 -l); and so on. It is easy to verify that the number of 
distinct three-sheeted surfaces, with 4 assigned points as simple branch-points, is 4 : an example 
suggested to me by Mr Burnside. 

t The continuation indicated will be carried out for the present case by means of the com 
bination of areas ( 222), and without further reference to the transformation indicated or to 
Schwarz s theorem on symmetrical continuation. 



213.] POTENTIAL FUNCTION 407 

The derivatives of u can then be obtained in the extended space and so v can 
be determined for the boundary*. 

And, what is more important, it will be found that, under conditions to be 
assigned, the number of functions u that are determined is double the number 
of functions w that are determined ; the complete set of functions u lead to all 
the parts u and v of the functions w ( 234, note). 

214. The infinities of u at any point are given by the real parts of the 
terms which indicate the infinities of w. Conversely, when the infinities of u 
are assigned in functional form, those of w can be deduced, the form of the 
associated infinities of v first being constructed by quadratures. 

The periods of w, being the moduli at the cross-cuts, lead to real constants 
as differences of u at opposite edges of cross-cuts, or, if we choose, as constant 
differences of values of u at points on definite curves, conveniently taken for 
reference as lines of possible cross-cuts. Conversely, a real constant modulus 
for u is the real part"f* of the corresponding modulus of w. 

Hence a function, w, of position on a Riemann s surface is, except as to an 

additive constant, determined by a real function u of x and y (where x + iy is 

the independent variable for the surface), if u be subject to the conditions : 

(i) it satisfies the equation V 2 u = at all points on the surface where 

its derivatives are not infinite : 

(ii) if it be multiform, its values at any point on the surface differ by 
linear combinations of integral multiples of real constants : otherwise, it is 
uniform : 

(iii) it may have specified infinities, of given form in the vicinity of 
assigned points on the surface. 

In addition to these general conditions imposed upon the function u, it is 
convenient to admit as a further possible condition, for portions of the surface, 
that the function u shall assume, along a closed curve, values which are 
always finite. But it must be understood that this condition is used only for 
subsidiary purposes : it will be seen that it causes no limitation on the final 
result, all that is essential in its limitations being merged in the three 
dominant conditions. 

The questions for discussion are therefore (i), the existence of functions J 
satisfying the above conditions in connection with a given Riemann s 

* See Phragmen, Acta Math., t. xiv, (1890), pp. 225227, for some remarks upon this 
question. 

t The imaginary parts of the moduli of w are determinate with the imaginary part of w : see 
remark at end of 213, and the further reference there given. 

The functions u (and also v) are of great importance in mathematical physics for two- 
dimensional phenomena in branches such as gravitational attraction, electricity, hydrodynamics 
and heat. In all of them, the function represents a potential ; and, consequently, in the general 
theory of functions, it is often called a potential function. 



408 METHODS OF SOLUTION [214. 

surface, the connectivity of which is 2p + 1 as dependent upon its branching 
and the number of its sheets; and (ii), assuming that the functions exist, 
their determination by the assigned conditions. 

215. There are many methods for the discussion of these questions. The 
potential function, both for two and for three dimensions in space, first arose in 
investigations connected with mathematical physics : and, so far as concerns 
such subjects, its theory was developed by Poisson, Green, Gauss, Stokes, 
Thomson, Maxwell and others. Their investigations have reference to appli 
cations to mathematical physics, and they do not tend towards the solution of 
the questions just propounded in relation to the general theory of functions. 

Klein uses considerations drawn from mathematical and experimental 
physics to establish the existence of potential functions under the assigned 
conditions. The proof that will be adopted brings the stages of the investi 
gation into closer relations with the preceding and the succeeding parts of the 
subject than is possible if Klein s method be followed*. 

To establish the existence of the functions under the assigned conditions, 
Riemann*f uses the so-called Dirichlet s Principle J ; but as Riemann s proof 
of the principle is inadequate, his proof of the existence-theorem cannot be 
considered complete. 

There are two other principal, and independent, methods of importance, 
each of which effectively establishes the existence of the functions, due to 
Neumann and to Schwarz respectively ; each of them avowedly dispenses^ 
with the use of Dirichlet s Principle. 

The courses of the methods have considerable similarity. Both begin 
with the construction of the function for a circular area. Neumann uses 
what is commonly called the method of the arithmetic mean, for gradual 
approximation to the value of the potential function for a region bounded 
by a convex curve : Schwarz uses the method of conformal representation, 
to deduce from results previously obtained, the potential function for 
regions bounded by analytical curves ; and both authors use certain 
methods for combination of areas, for each of which the potential function 
has been constructed ||. 

* Klein s proof occurs in his tract, already quoted, Ueber Riemann s Theorie der algebraischen 
Functionen und Hirer Integrate, (Leipzig, Teubner, 1882), and it is modified in his memoir "Neue 
Beitrage zur Eiemann schen Functionentheorie, " Math. Ann., t. xxi, (1883), pp. 141 218, 
particularly pp. 160 162. 

t Ges. Werke, pp. 3539, pp. 9698. 

J Eiemann enunciates it, (I.e.), pp. 34, 92. 

Neumann, Vorlesungen liber Riemann s Theorie der Abel schen Integrale, (2nd ed., 1884), 
p. 238; Schwarz, Ges. Werke, ii, p. 171. 

|| Neumann s investigations are contained in various memoirs, Math. Ann., t. iii, (1871), 
pp. 325349; ib., t. xi, (1877), pp. 558566; ib., t. xiii, (1878), pp. 255300; ib., t. xvi, 
(1880), pp. 409 431 ; and the methods are developed in detail and amplified in his treatise 






215.] SUMMARY OF SCHWARZ S ARGUMENT 409 

What follows in the present chapter is based upon Schwarz s investi 
gations : the next chapter is based upon the investigations of both Schwarz 
and Neumann, and, of course, upon Riemann s memoirs. 

The following summary of the general argument will serve to indicate the main line of 
the proof of the establishment of potential functions satisfying assigned conditions. 

I. A potential function u is uniquely determined by the conditions : that it, as 

du du d 2 u c) 2 u , , . , . ., , . -. , ,, i 

well as its derivatives -, ~-, 5-3, ^ 2 - (which satisfy the equation V%=0), shall be 

uniform, finite and continuous, for all points within the area of a circle ; and that, along 
the circumference of the circle, the function shall assume assigned values that are always 
finite, uniform and, except at a limited number of isolated points where there is a sudden 
(finite) change of value, continuous. ( 216 220.) 

II. By using the principle of conformal representation, areas bounded by curves other 
than circles say by analytical curves are obtained, over which the potential function is 
uniquely determined by general conditions within the area and assigned values along its 
boundary. ( 221.) 

III. The method of combination of areas, dependent upon an alternating process, 
leads to the result that a function exists for a given region, satisfying the general conditions 
in that region and acquiring assigned finite values along the boundary, when the region 
can be obtained by combinations of areas that can be conformally represented upon the 
area of a circle. ( 222.) 

IV. The theorem is still valid when the region (supposed simply connected) contains 
a branch-point ; the winding-surface is transformed by a relation 

z-c = RZ m 
into a single-sheeted surface, for which the theorem has already been established. 

When the surface is multiply connected, we resolve it by cross-cuts into one that is 
simply connected, before discussing the function. ( 223.) 

V. Real functions exist on a Riemann s surface, which are everywhere finite and 

Ueber das logarithmische und Newton sche Potential (Leipzig, Teubner, 1877) and in his treatise 
quoted in the preceding note. In this connection, as well as in relation to Schwarz s investi 
gations, and also in view of some independence of treatment, Harnack s treatise, Die Grundlagen 
der Theorie des logarithmischen Potentiates und der eindeutigen Potentialfunction in der Ebene 
(Leipzig, Teubner, 1887), and a memoir by Harnack, Math. Ann., t. xxxv, (1890), pp. 1940, 
may be consulted. 

A modification of Neumann s proof, due to Klein, is given in the first volume (pp. 508 522) 
of the treatise cited on p. 403, note. 

Schwarz s investigations are contained in various memoirs occurring in the second volume 
of his Gcsammclte Werke, pp. 108132, 133143, 144171, 175210, 303306 : their various 
dates and places of publication are there stated. A simple and interesting general statement 
of the gist of his results will be found in a critical notice of the two volumes of his collected 
works, written by Henrici in Nature (Feb. 5, 12, 1891, pp. 321323, 349352). There is a 
comprehensive memoir by Ascoli, based upon Schwarz s method, " Integration der Differential- 
gleichung V 2 w = in einer beliebigen Riemann schen Fla che," (Bih. t. kongl. Svenska Vet. Akad. 
Handl., bd. xiii, 1887, Afd. 1, n. 2 ; 83 pp.) ; a thesis by Jules Riemann, Sur le probleme de 
Dirichlet, (These, Gauthier-Villars, Paris, 1888), discusses a number of Schwarz s theorems 
(see, however, Schwarz, Ges. Werke, t. ii, pp. 356358) ; and an independent memoir by Prym, 
Crelle, t. Ixxiii, (1871), pp. 340364, may be consulted. 

The literature of this part of the subject is very wide in extent : many other references are 
given by the authors already quoted. 



410 PRELIMINARY LEMMAS [215. 

uniquely determinate by arbitrarily assigned real moduli of periodicity at the cross-cuts. 
( 224227.) 

VI. Functions exist, satisfying the conditions in (V) except that they may have at 
isolated points on the surface, infinities of an assigned form. ( 229.) 

216. We shall, in the first place, treat of potential functions that have 
110 infinities, either algebraical or logarithmic, over some continuous area on 
the surface limited by a simple closed boundary, or by a number of non-inter 
secting simple closed curves constituting the boundary ; for the present, the 
area thus enclosed will be supposed to lie in one and the same sheet, so that 
we may regard the area as lying in a simple plane. 

At all points within the area and on its boundary, the function u is finite 
and will be supposed uniform and continuous ; for all points within the area 
(but not necessarily for points on the boundary), the derivatives 

du du d 2 u d 2 u 
dx dy dx* dy 2 

are uniform, finite arid continuous and they satisfy the equation V 2 it = 0. 
These may be called the general conditions. 

Two cases occur according as the character of the derivatives at points in 
the area is or is not assigned for points on the boundary ; if the character be 
assigned, there will then be what may be called boundary conditions. The 
two cases therefore are : 

(A) When a function u is required to satisfy the general conditions, 
and its derivatives are required to satisfy the boundary conditions : 

(B) When the only requirement is that the function shall satisfy the 
general conditions. 

Before proceeding to the establishment of what is the fundamental 
proposition in Schwarz s method, it is convenient to prove three lemmas 
and to deduce some inferences that will be useful. 

LEMMA I. If two functions u and u 2 satisfy the general conditions for two 
regions T : and T 2 respectively, which have a common portion T that is more than 
a point or a line, and if u t and u 2 be the same for the common portion T, then 
they define a single function for the whole region composed of T : and T 2 . 

This proposition can be made to depend upon the continuation of 
analytical functions*, whether in a plane ( 34) or, in view of a subsequent 
transformation ( 223), on a Biemann s surface. 

The real function u r defines a function Wj_ of the complex variable z, for any 
point in the region T^ ; and for points within this region, the function w 1 is 
uniquely determined by means of its own value and the values of its deriva 
tives at any point within T 1} obtained, if necessary, by a succession of elements 
* For other proofs, see Schwarz, ii, pp. 201, 202 and references there given. 



216.] FOR SCHWARZ S PROOF 411 

in continuation. Hence the value of w l and its derivatives at any point 
within T defines a function existing over the whole of T^. 

Similarly the real function u 2 defines a function w. 2 within T 2 , and this 
function is uniquely determined over the whole of T 2 by its value and the 
value of its derivatives at any point within T. 

Now the values of u^ and u 2 are the same at all points in T, and therefore 
the values of w l and w 2 are the same at all points in T, except possibly for an 
additive (imaginary) constant, say ia, so that 

w-i = w 2 + ia. 

Hence for all points in T, (supposed not to be a point, so that we may have 
derivatives in every direction ( 8) : and not to be a line, so that we may 
have derivatives in all directions from a point on the line), the derivatives 
of w l agree with those of w. 2 ; and therefore the quantities necessary to define 
the continuation of w l from T over Tj agree with the quantities necessary to 
define the continuation of w 2 from T over T 2 , except only that w l and w 2 
differ by an imaginary constant. Hence, having regard to the form of the 
elements, w { and w. 2 can be continued over the region composed of T l and T 2 , 
and their values differ (possibly) by the imaginary constant. When we take 
the real parts of the functions, we have u t and u 2 defining a single function 
existing over the whole region occupied by the combination of T and T 2 . 

The other two lemmas relate to integrals connected with potential 
functions. 

LEMMA II. Let u be a function required to satisfy the general conditions, 
and let its derivatives be required to satisfy the boundary conditions, for an 
area S bounded by simple non-intersecting curves : then 

du , .. 
5- ds = : 
on 

where the integral is extended round the whole boundary in the direction that is 
positive with regard to the bounded area 8 ; and dn is an element of the normal 
to a boundary-line drawn towards the interior of the space enclosed by that 
boundary-line regarded merely as a simple closed curve*. 

Let P and Q be any two functions, which, as well as their first and second 
derivatives with regard to # and to y, are uniform finite and continuous for 
all points within S and on its boundary. Then, proceeding as in 16 and 
taking account of the conditions to which P and Q are subject, we have 



/Yp\7-r)7 ,7 fvfiQj d $j\ [[f dp dQ , dP3Q\j j 
PV-Qdxdy = P (~ dy - ~ dx }- U- + -^ ~ dacdy ; 

Jj J \<Mt dy ) JJ\dx dx oydyj 



dy ) JJ\dx dx oydy 

* The element dn of the normal is, by this definition, measured inwards to, or outwards 
from, the area S according as the particular boundary-line is described in the positive, or in the 
negative, trigonometrical sense. Thus, if S be the space between two concentric circles, the 
element dn at each circumference is drawn towards its centre ; the directions of integration are 
as in 2. 





412 PRELIMINARY LEMMAS [216. 

3 2 3 2 
where V 2 denotes = h ^-- , the double integrals extend over the area of S, and 

dx 2 dy 2 

the single integral is taken round the whole boundary of S in the direction 
that is positive for the bounded area S. 

Let ds be an element PT of arc of the boundary at a point (x, y), and dn be 
an element TQ of the normal at T drawn to the 
interior of the space included by the boundary- 
line regarded as a simple closed curve ; and let ty 
be the inclination of the tangent at T. Then in 
(i), as TQ is drawn to the interior of the area in- P p 

eluded by the curve, the direction of integration lg- 78- 

being indicated by the arrow (so that S lies within the curve), we have 

dx = ds cos ^r dn sin ty, dy = ds sin ty + dn cos i|r ; 
and therefore it follows that, for any function R, 

dR dR . dR 

tr = TT Sin W + -^ COS -vjr. 

dn dx dy 

Now for variations along the boundary we have dn = 0, so that 

dR,_dRj 3E , 

~"~ ~^r CvO /* CvtJ ~~* *" \JjvU, 

on ox oy 

And in (ii), as TQ is drawn to the interior of the area included by the curve, 
the direction of integration being indicated by the arrow (so that 5 lies 
without the curve), we have 

dx = ( ds) cos -vjr + dn sin -fr, dy ( ds) sin -v/r dn cos ty, 

dR dR . dR 

and therefore -^ = = sin y ^ cos Y, 

dn ox oy 

so that, as before, for variations along the boundary, 

dR . dR . dR , 
^- ds = ^ dy -=- dx. 
on ox oy 

Hence, with the conventions as to the measurement of dn and ds, we have 



both integrals being taken round the whole boundary of S in a direction that 
is positive as regards S. Therefore 

- - f P * * - \i(^ % + 3 / 
J on JJ\ox dx dy oy 

In the same way, we obtain the equation 



f[nv vj i fn 

QV 2 Pdxdy = - IQ 
JJ J 



a 

dn JJ\dx dx dy dy 

and therefore (PV 2 Q - QV P) d^?y = ff Q |? - P ^) cfo, 

y 



216.] FOR SCHWARZ S PROOF 413 

where the double integral extends over the whole of 8, and the single 
integral is taken round the whole boundary of S in the direction that is 
positive for the bounded area S. 

Now let u be a potential function defined as in the lemma; then u 
satisfies all the conditions imposed on P, as well as the condition V 2 w = 
throughout the area and on the boundary. Let Q = I ; so that V 2 Q 0, 

= 0. Each element of the left-hand side is zero, and there is no dis- 

dn 

continuity in the values of P and Q ; the double integral therefore vanishes, 
and we have 

f 5- & =s 0, 

J dn 
the result Avhich was to be proved. 

But if the derivatives of u are not required to satisfy the boundary 
conditions, the foregoing equation may not be inferred ; we then have the 
following proposition. 

LEMMA III. Let u be a function, which is only required to satisfy the 
general conditions for an area S ; and let u be any other function, which 
is required to satisfy the general conditions for that area and may or may 
not be required to satisfy the boundary conditions. Let A be an area entirely 
enclosed in S and such that no point of its whole boundary lies on any part of 
the whole boundary of S ; then 



|V du f ,du\, 

(u = u 5- )ds = 0, 

J\ dn on) 



where the integral is taken round the whole boundary of A in a direction 
which is positive with regard to the bounded area A, and the element dn of 
the normal to a boundary-line is drawn towards the interior of the space 
enclosed by that boundary-line, regarded merely as a simple closed curve. 

The area A is one over which the functions u and u satisfy the general 
conditions. The derivatives of these functions satisfy the boundary-conditions 
for A, because they are uniform, finite and continuous for all points inside S, 
and the boundary of A is limited to lie entirely within S. Hence 



the integrals respectively referring to the area of A and its boundary in a 
direction positive as regards A. But, for every point of the area, V 2 w = 0, 
W = ; and u and u are finite. Hence the double integral vanishes, and 
therefore 



(f 

l( 
J\ 



du ,du 

o -- u ^- 
dn dn 



taken round the whole boundary of A in the positive direction. 



414 POTENTIAL FUNCTION [216. 

One of the most effective modes of choosing a region A of the above 
character is as follows. Let a simple curve (7j be drawn lying entirely within 
the area S, so that it does not meet the boundary of 8; and let another 
simple curve C 2 be drawn lying entirely within C l , so that it does not meet 
(7j and that the space between C\ and (7 2 lies in S. This space is an area of 
the character of A, and it is such that for all internal points, as well as for 
all points on the whole of its boundary (which is constituted by C^ and (7 2 ), 
the conditions of the preceding lemma apply. The curve (7 2 in the above 
integration is described positively relative to the area which it includes : the 
curve C-i is described, as in 2, negatively relative to the area which it 
includes. Hence, for such a space, the above equation is 

// du , du\ , (( du , du\ 

MM 5 u 5- ] dSi - nu -= u 5- }ds 2 = 0, 

J\ dn Cn) J\ dn dnj 

if the integrals be now extended round the two curves in a direction that is 
positive relative to the area enclosed by each, and if in each case the normal 
element dn be drawn from the curve towards the interior. 

217. We now proceed to prove that a function u, required to satisfy the 
general conditions for an area included within a circle, is uniquely determined 
by the series of values assigned to u along the circumference of the circle. 

Let the circle 8 be of radius R and centre the origin. Take an internal 
point z = reP 1 , and its inverse z = r e^ (such that rr f = R) : so that z is 
external to the circle. Then the curves determined by 



Zr, 



for real values of X, are circles which do not meet one another. The boundary 
of 8 is determined by X = 1, and X = gives the point z as a limiting circle : 
and the whole area of S is obtained by making the real parameter X 
change continuously from to 1. 

Lemma III. may be applied. We choose, as the ring-space, the area 
included between the two circles determined by Xj and X 2 , where 

1 > Xj > X, > ; 
and then we have 



[( du ,du\ 1 (I du 
llu -^ -- u \ds 1 = l(u ^ -- u 
J \ on dnj J \ dn 



,du\ 
5- 

on] 



where the integrals are taken round the two circumferences in the trigono- 
metrically positive direction (dn being in each case a normal element drawn 
towards the centre of its own circle), and the function u satisfies the general 
and the boundary conditions for the ring-area considered. Moreover, the 
area between the circles, determined by \ and X 2 , is one for which u satisfies 



217.] DETERMINED FOR A CIRCLE 415 

the general conditions, and its derivatives certainly satisfy the boundary 
conditions : hence 



fdu .. fdu 

U- cfej = 0, 1 5- ds 2 = 0. 

J dn J dn 

Now the function u is at our disposal, subject to the general conditions 
for the area between the two X-circles and the boundary conditions for each 
of those circles. All these conditions are satisfied by taking u as the real 

(z z \ 
-, j , that is, in the present case, 
Z ZQ / 

, , I * - Ji 
u =log z ~-^, . 

(T \ 
P Xj j , so 

that 

u ^ ds l = : 

J dn 

and similarly for all points on the inner circle u is equal to the constant 

/ r \ 

log I p X 2 ) , so that 
\zi / 

/ |>=o. 

Again, for a point z on the outer circle, whose angular coordinate is ^r, 

du 

the value of ^ for an inward drawn normal is (5 11) 
dn 

(E 2 -r 2 V) 2 



_ 
\,R (R 2 - r 2 ) {R 2 - 2Rr\ cos (^ - 0) + r 2 ^ 2 } 

and because the radius of that outer circle is \R (R- r 2 )/(R 2 r 2 \ 1 2 ), we 
have 

, \ 1 R(R 2 -r 2 ), 



Denoting by/(X!, x|r) the value of u at this point i/r on the circle determined 
by Xj, we have 



(- + ^v 

Similarly for the inner circle, the normal element again being drawn towards 
its centre, we have 



Combining these results, we have 

2 ^ 2 



T 27 
Jo 



416 INTEGRAL-EXPRESSION [217. 

In the analysis which has established this equation, Xj and X 2 can have all 
values between 1 and : the limiting value is excluded because then u 
is not finite, and the limiting value 1 is excluded because no supposition has 
been made as to the character of the derivatives of u at the circumference 
of 5. 

The equation which has been obtained involves only the values of u 
but not the values of its derivatives. Since the values of u are finite both 
for X = and \ = 1, and the integrals are finite, the exclusion of the limiting 
values of X need not be applied to the equation, although the exclusion was 
necessary during the proof, owing to the presence of quantities that have 
since disappeared. Hence the equation is valid when we take \ = 1, X 2 = 0. 

When X*, 0, the corresponding circle collapses to the point z : the value 
of y(X 2 , ty) is then the value of u at # say u(r, <); and the integral 
connected with the second circle is ZTTU (r, <). 

When Xj = 1, the corresponding circle is the circle of radius R ; the value 
of /(Xj, ty) is then the assigned value of u at the point i/r on the circum 
ference, say the function /(^). Substituting these values, we have 

u ^ V=L 



the integral being taken positively round the circumference of the circle 8. 

It therefore appears that the function u, subjected to the general 
conditions for the area of the circle, is uniquely determined by the values 
assigned to it along the circumference of the circle. 

The general conditions for u imply certain restrictions on the boundary 
values. These values must be finite, continuous and uniform : arid therefore 
as a function of ty, must be finite, continuous, uniform and periodic in 
of period 2-7T. 



218. It is easy to verify that, when the boundary values f(ty) are not 
otherwise restricted, all the conditions attaching to u are satisfied by the 
function which the integral represents. 

Since the real part of (Re^ + z)j(Re^ - z) is the fraction 

(R> - r 2 )/{E 2 - 2Rr cos (^ - <) + r 2 }, 
it follows that u is the real part of the function F (z), defined by the equation 



. . 

F( z \= ^-77 
^ l 



For all values of z such that z < R, the fraction can be expanded in a series 
of positive integral powers of z, which converges unconditionally and uni 
formly ; and therefore F (z) is a uniform, continuous, analytical function, 



218.] FOR A POTENTIAL FUNCTION 417 

everywhere finite for such values of z. Hence all its derivatives are uniform, 
continuous, analytical functions, finite for those values of z\ and these 
properties are possessed by the real and the imaginary parts of such 

d m + n U (frn+n p / z \ 

derivatives. Now ^-^ is the real part of ^.-jL-Zj and therefore, 

for all integers m and n positive or zero, it is a uniform, finite and continuous 
function for points such that z\<R, that is, for points within the circle. 
Moreover, since u is the real part of a function of z, and has its differential 
coefficients uniform, finite and continuous, it satisfies the differential equation 
V"u = 0. 

To infer the continuity of approach of u(r, <) to /(<) as r is made equal 
to R, we change the integral expression for u (r, <) into 



Moreover for all values of r < R (but not for r = R), we have 

1 /2"--< ft 2 r 2 If (7?4-r 

^ P^?- ~l^ de = ^n- HTtani =1; 

%TT J -$ R 2 2Rr cos V -f r" TT [_ (R r 



and therefore 

/-(.+)-/<+) 

7?2 



Let denote the subject of integration in the last integral. Then, as r 
is made to approach indefinitely near to R in value, becomes infinitesimal 
for all values of 6 except those which are extremely small, say for values of 
between - $ and + 8. Dividing the integral into the corresponding parts, 
we have 



Let M be the greatest value of /(i/r) for points along the circle. Then the 
first integral and the second integral are less than 



_ j 

27T - a 



respectively ; by taking r indefinitely near to R in value, these quantities 
can be made as small as we please. For the third integral, let k be the 
greatest value of f(<f> + 6) -/(</>) for values of 6 between 8 and - 8 : then the 
third integral is less than 

k f s R?-r* 

2-7T J _ 5 R*-* 



that is, it is less than - tan" 1 (^ -^~ j so that, when r is made nearly 
equal to R, the third integral is less than k. 

F 27 



418 DISCONTINUITY IN VALUE [218. 

If then k be infinitesimal, as is the case when /((/>) is everywhere finite 
and continuous, the quantity / can be diminished indefinitely; hence u(r, (/>) 
continuously changes into the function /(<) as r is made equal to R. The 
verification that the function, defined by the integral, does satisfy the general 
conditions for the area of the circle and assumes the assigned values along 
the circumference is thus complete. 

Ex. Shew that, if M denote the maximum value (supposed positive) of/(^) for points 
along the circumference of the circle and if u (0) denote the value of the function at the 
centre, then 



I u (r, d>) - u (0) | < - M sin- 1 



also that, if u (0) vanish, then 

u (r, $)< - Mtan~ l ^ . (Schwarz.) 



219. But in view of subsequent investigations, it is important to consider 
the function represented by the integral when the periodic function /(</>) 
which occurs therein is not continuous, though still finite, for all points on 
the circumference. The contemplated modification in the continuity is that 
which is caused by a sudden change in value of /(<) as <j> passes through a] 
value a : we shall have 

/( + )-/(-)- -A, 
when e is ultimately zero. Then the following proposition holds : 

Let a function f($) be periodic in ZTT, finite everywhere along the circle, j 
and continuous save at an assigned point a where it undergoes a sudden increase 
in value : a function u can be obtained, which satisfies the general conditions 
for the circle except at such a point of discontinuity in the value of f((f>\ and 
acquires the values of f((f>) along the circumference. 

Let p be a quantity < R : then along the circumference of a circle of radius 
p, the general conditions are everywhere satisfied for the function u, so that, if 
u (p, i/r) be the value at any point of its circumference, the value of u at any 
internal point is given by 

u (r, $} = ~ u (p, W p2 _ 2/5rc P os " ( ^_ </)) + r2 <** 

Now p can be gradually increased towards R, because the general conditions 
are satisfied; but, when p is actually equal to R, the continuity of 
u(p, ^r) is affected at the point a. We therefore divide the integral into 
three parts, viz., to a - e, a - e to a + e, and a + e to 2-n-, when p is very 
nearly equal to R. For the first and the third of these parts, p can, as in the 
preceding investigation, be changed continuously into R without affecting 
the value of the integral. If we denote by p the integral 

~ 



where the range of integration does not include the part from a e to a + e, 



219.] ALONG THE CIRCUMFERENCE 419 

and where the values /(a e), /(a + e) are assigned to u (R, a e), u (R, a. + e), 
respectively ; the sum of the integrals for the first and the third intervals is 
p + A, where A is a quantity that vanishes with R p, because the subject of 
integration is everywhere finite. For the second interval, the integral is 
equal to q + A , where 

1 [ a+f R 2 r 2 



and A is a quantity vanishing with R p because the subject of integration 
is everywhere finite. So far as concerns q, let M be the greatest value of 

l/Wh ^en 

, MR+r 

l?l< 2ir.B-r 

a quantity which, because M is finite (but only if M be finite), can be made 
infinitesimal with e, provided r is never actually equal to R. If then, an 
infinitesimal arc from a e to a + e be drawn so as, except at its assigned 
extremities, to lie within the area of the circle, the last proviso is satisfied : 
and the effect is practically to exclude the point a from the region of 
variation of u as a point for which the function is not precisely defined. 
With this convention, we therefore have 

2?r 



_ ~ 



so that, by making p ultimately equal to R and e as small as we please, the 
difference between u (r, <) and the integral defined as above can be made zero. 
Hence the integral is, as before, equal to the function u (r, 0), provided that 
the point a be excluded from the range of integration, the value /(a e) just 
before ^=0. and the value /(a+e) just after ^=0. being assigned to u (R, ^r). 
It therefore appears that discontinuities may occur in the boundary 
values when the change is a finite change at a point, provided that all 
the values assigned to the boundary function be finite. 

COROLLARY. The boundary value may have any limited number of points 
of discontinuity, provided that no value of the function be infinite and that at 
all points other than those of discontinuity the periodic function be uniform, 
finite and continuous : and the integral will then represent a potential function 
satisfying the general conditions. 

The above analysis indicates why discontinuities, in the form of infinite 
values at the boundary, must be excluded: for, in the vicinity of such a 
point, the quantity M can have an infinite value and the corresponding 
integral does not then necessarily vanish. Hence, for example, the real 
part of 



is not a function that, under the assigned conditions, can be made a boundary 
value for the function u. 



272 




420 SPECIAL FORMS [219. 

It is easy to construct a function with permissible discontinuities. We know ( 3) 
that the argument of a point experiences a sudden change by p 

TT when the path of the point passes through the origin. Let 
a point P on a circle be considered relative to A : the inclina- Q / 
tion of AP to the normal, drawn inwards at A, is - - (o - <), 

and of A Q to the same line is - | - |(a - ) , so that there 

is a sudden change by TT in that inclination. Now, taking a function 

A^ ,r fir i/ ,01 
# W>)= tan- 1 tan \=- f (a- <p}\ I, 

7T L. {^ J J 

and limiting the angle, defined by the inverse function, so that it lies between - \n 
and +^TT, as may be done in the above case and as is justifiable with an argument 
determined inversely by its tangent, the function g (0) undergoes a sudden change A as <p 
increases through the value a. Moreover, all the values of g (<) are finite : hence g (<) is 
a function which can be made a boundary value for the function u. Let the function 
thence determined be denoted by u a . 

By means of the functions u a , we can express the value of a function u whose boundary 
value /(<) has a limited number of permissible discontinuities. Let the increases in value 
be A lt ...,A m at the points a l5 a 2 , ... , a TO respectively : then, if g n (<j>) denote 



we have g n (a n + f}-g n ( - * ) = A n, when e is infinitesimal. Hence 

has no discontinuity at a n , that is,/(0)-^(0) has no discontinuity at a n . 

Hence also /(<)- 2 #(<) has no discontinuity at a 1? ..., a m , and therefore it is 

n~ 1 

uniform, finite, and continuous everywhere along the circle; and it is periodic in 27r. 
By 218, it determines a function U which satisfies the general conditions. 

Each of the functions g n (0) determines a function u n satisfying the general conditions : 
hence, as u is determined by /(<), we have 

m 

U- S U n =U, 
n=l 

which gives an expression for u in terms of the simpler functions u n and of a function U 
determined by simpler conditions as in 218. 

Ex. Shew that, if /(\|^) = 1 from - \ir to +\n and =0 from +^TT to f TT, then is the 
real part of the function 

1. 1+12 

log -.- 

^7^ " 



The general inference from the investigation therefore is, that a function 
of two real variables x and y is uniquely determined for all points within a 
circle by the following conditions : 

(i) at all points within the circle, the function u and its derivatives 

du du &u &u be unifo finite anc } continuous, and 

dx dy dx* 8y 2 

must satisfy the equation V 2 i* = : 

(ii) if /(() denote a function, which is periodic in </> of period 2?r, is 
finite everywhere as the point <f> moves along the circumference, 



219.] GENERAL PROPERTIES 421 

is continuous and uniform at all except a limited number of 
isolated points on the circle, and at those excepted points 
undergoes a sudden prescribed (finite) change of value, then 
to u is assigned the value /(<) at all points on the circumference 
except at the limited number of points of discontinuity of that 
boundary function. 

And an analytical expression has been obtained, the function represented by 

which has been verified to satisfy the above conditions. 

220. We now proceed to obtain some important results relating to a 
function u, defined by the preceding conditions. 

I. The value of u at the centre of the circle is the arithmetic mean of its 
values along the circumference. 
For, by taking r = 0, we have 



the right-hand side being the arithmetic mean along the circumference. 

II. If the function be a uniform constant along the circumference, it is 
equal to that constant everywhere in the interior. 

For, let C denote the uniform constant ; then 



= 
for all values of r less than R, that is, everywhere in the interior. 

But if the function, though not varying continuously along the circum 
ference, should have different constant values in different finite parts, as, for 
instance, in the example in 219, then the inference can no longer be drawn. 

III. If the function be uniform, finite and continuous everywhere in the 
plane, it is a constant. 

Since the function is everywhere uniform, finite and continuous, the 
radius R of the circle of definition can be made infinitely large : then, as 
the limit of the fraction (R 2 r-)j{R 2 2Rr cos (ty () + r 2 } is unity, we 

have 

1 f 2 " 
u(r, <)=2^.J w(co,xjr)cty, 

the integral being taken round a circle of infinite radius whose centre is the 
origin. But, by (I) above, the right-hand integral is u (0), the value at the 
centre of the circle ; so that 

u.(r, <j>) = u(Q), 

and therefore u has the same value everywhere. 

This is practically a verification of the proposition in 40, that a uniform, 
finite and continuous function w, which has no infinity anywhere, is a constant. 



422 GENERAL PROPERTIES [220. 

IV. A uniform, finite and continuous function u cannot have a maximum 
value or a minimum value at any point in the interior of a region over which,, 
subject to the general conditions as to the differential coefficients, it satisfies the 
differential equation V 2 M = 0. 

If there be any such point not on the boundary, it can be surrounded by 
an infinitesimal circle for the interior of which, as well as for the circum 
ference of which, u satisfies both the general and the boundary conditions ; hence 



[du 7 A 
I ~- ds = 0, 

J on 



the integral being taken round the circumference. But in the immediate 

p 

vicinity of such a point, ~- has everywhere the same sign, so that the 
integral cannot vanish : hence there is no such point in the interior. 

In the same way, it may be proved that there cannot be a line of 
maximum value or a line of minimum value within the surface : and that 
there cannot be an area of maximum value or an area of minimum value 
within the surface. 

V. It therefore follows that the maximum values for any region are to be 
found on its boundary : and so also are the minimum values. 

If M be the maximum value, and if m be the minimum value of the 
function for points along the boundary, then the value of the function for an 
interior point is < M and is > m and can therefore be represented in the form 
Mp + m (1 p), where p is a real positive proper fraction, varying from point 
to point. 

In particular, let a function have the value zero for a part of the 
boundary and have the value unity for the rest : the value that it has for 
points along a line in the interior is always positive and has an upper limit 
q, a proper fraction. But q will vary from one line to another. If the region 
be a circle and q be the proper fraction for a line in the circle, then the value 
along that line of a function u, which is still zero over the former part of the 
boundary but has a varying positive value ^. p along the remainder, is 
evidently ^ qp. This fraction q may be called the fractional factor for the 
line in the supposed distribution of boundary values. 

VI. It may be noted that the second of these propositions can now 
be deduced for any simply connected surface. For when a function is 
constant along the boundary, its maximum value and its minimum value 
are the same, say X: then its value at any point in the interior is 
\p + \(l p), that is, X, the same as at -the boundary. Consequently if 
two functions % and u 2 satisfy the general conditions over any region, and 
if they have the same value at all points along the boundary, then they 
are the same for all points of the region. For their difference satisfies 



220.] EXISTENCE FOR ANALYTICAL CURVE 423 

the general conditions : it is zero everywhere along the boundary : hence 
it is zero over the whole of the bounded region. 

If, then, a function u satisfy the general conditions for any region, it is 
unique for assigned boundary values that are everywhere finite, uniform, and 
continuous except at isolated points. 

221. The explicit expression of u with boundary values, that are 
arbitrary within the assigned limits, has been determined for the area 
enclosed by a circle : the determination being partially dependent upon the 
form assumed in 217 for the subsidiary function u . The assumption of 
other forms for u , leading to other curves dependent upon a parametric 
constant, would lead by a similar process to the determination of u for the 
area limited by such families of curves. 

But without entering into the details of such alternative forms for u , we 
can determine the value of u, under corresponding conditions, for curves 
derivable from the circle by the principle of conformal representation*. 
Suppose that, by means of a relation 



or, say x + iy = p (f , 77) + iq (, 77), 

where p and q are real functions of and 77, the area contained within the 
circle is transformed, point by point, into the area contained within another 
curve which is the transformation of the circle : then the function u (x, y) 
becomes, after substitution for x and y in terms of and 77, a function, say U, 
of and 77. 

Owing to the character of the geometrical transformation, p and q (and 
their derivatives with regard to and 77) are uniform, finite and continuous 
within corresponding areas. Hence 



8 U _ du dp dudq 8 U _ du dp du dq _ 
8 dx dj; dy 8^ 8/7 dx 877 dy 877 

d*U d n -U (&u 8%\ (78/A 2 /8V 

and aS" + -5-5 = a"* + 51 1 U6- + a 

dp 877 2 \dx 2 dy 2 J \\di-J \drjj 

so that the function U satisfies the general conditions for the new area 
bounded by the new curve. 

Moreover, u has assigned values along the circular boundary which is 
transformed, point by point, into the new boundary ; hence U has those 
assigned values at the corresponding points along the new boundary. Thus 
the function U is uniquely determined for the new area by conditions which 
are exactly similar to those that determine u for a circle : and therefore the 

* The general idea of the principle, and some illustrations of it, as expounded in 
Chapters XIX and XX, will be assumed known in the argument which follows : see especially 
265, 266. 



424 POTENTIAL FUNCTION [221. 

potential function is uniquely determined for any area, which can be con- 
formally represented on the area of a circle, by the general conditions of 
216 and the assignment of values that are finite and, except at a limited 
number of isolated points where they may suffer sudden (finite) changes of 
value, uniform and continuous at all points along the boundary of the area. 

One or two examples of very special cases are given, merely by way of 
illustration. The general theory of the transformation of a circle or an 
infinite straight line into an analytical curve will be considered in Chapter 
XX. But, meanwhile, it is sufficient to indicate that, by the principle of 
conformal representation, we can pass from the circle to more general curves 
as the boundary of an area within which the potential function is defined by 
conditions similar to those for a circle : in particular that, by assuming the 
result of 265, 266, we can pass from the circle to an analytical curve as the 
boundary of such an area. 

Ex. 1. A function u satisfying the general conditions for a circle of radius unity and 
centre the origin, and having assigned values /(^r) along the circumference, is determined at 
any internal point by the equation 

" 



Now the circle and its interior are transformed by the equation 

+1-3 
r 

into a parabola and the excluded area (Ex. 7, 257) : so that, if R, 6 be polar coordinates 
of any point in that excluded area, we have 



Corresponding to the circle / = !, we have the parabola 

Rca&\6 = \; 
if determine the point on the parabola, which corresponds to \|/- on the circle, we have 



or TJr = Q. 

Hence the function U(R, 6) assumes the values /(9) along the boundary of the 
parabola. 

Also l-r 2 = ^(R*cos$d-l), 

1 - 2r cos ty - 0) + r 2 =-i [R cos 2 - 2/2* cos cos (e + 6) + 1] ; 

Mm 

and therefore we have the following result : 

A function ivhich satisfies the general conditions for the area bounded by and lying on the 
convex side of the parabola Rcos 2 ^Q = l and is required to assume the value /(6) at points 
along the parabola, is defined uniquely for a point (r, 6} external to the parabola by the 
integral 



The function /(O) may suffer finite discontinuities in value at isolated points: elsewhere 
it must be finite, continuous and uniform. 



221.] FOR CONFORMALLY RELATED AREAS 425 

Ex. 2. Obtain an expression for u at points within the area of the same parabola, by 
using 



as the equation of transformation of areas ( 257). 
Ex. 3. When the equation 



is used, then, if z = x+iy and =X+iY, we have 

. _ 

+ iy ~ 



If the point describe the whole length of the axis of X from - oo to + oo , so that we 
may take f=JT=tan^> with $ increasing from -\K to +%TT, we have #=cos2</>, 
y = sin20; and z describes the whole circumference of a circle, centre the origin and 
radius unity, in a trigonometrically positive direction beginning at the point ( - 1, 0). We 
easily find 

rcos0 rsind r 2 1 



where = RcosQ, r) = RsmQ. Moreover, for variations along the circumference, we 
have i|r = 2<; whence, substituting and denoting by F(x), =/(2 tanr 1 ;?), the value of 
the potential at a point on the axis of real quantities whose abscissa is x, we ultimately 
find 



as the value of the potential-function u at a point (R, Q) in the upper half of the plane, 
when it has assigned values F(x] at points along the axis of real variables. 

222. The function u has now been determined, by means of the general 
conditions within an area and the assigned boundary values, for each space 
obtained by the method indicated in 221. But the determination is 
unique and distinct for each space thus derived ; and, if two such spaces 
have a common part, there are distinct functions u. We now proceed to 
shew that when two spaces, for each of which alone a function u can be 
determined, have a common part which is not merely a point or a line, 
then the function u is uniquely determined for the combined area by the 
assignment of finite, uniform and continuous values {or partially discontinuous 
values, as in 219) along the boundary of the combined area. 

Let the spaces be 2\ and jP 2 having a 
common part T, so that the whole space 
can be taken in the form T l + T 2 - T. Let 
the part of the boundary of T^ without T^ 
be L , and the part within T 2 be L. 2 : and 
similarly, for the boundary of T 2 , let L^ de 
note the part within T a and L 3 the part 
without it. Then the boundary of 

I T 1+ T>-T Fig - 80 

is made up of L and L 3 : the boundary of T is made up of L : and L,. 




426 COMBINATION [222. 

With an assignment of zero value along L Q and unit value along L. 2 , 
let the fractional factor ( 220, V), for the line L in the region T^ be q ; 
and with an assignment of zero value along L s and unit value along L lt let 
the fractional factor along the line L 2 in the region T z be q 2 . Then q 1 and q z 
are positive proper fractions. 

Let any series of values be assigned along L and L 3 subject to the 
conditions of being uniform, finite everywhere, and discontinuous, if at all, 
only at a limited number of isolated points ; these values are the boundary 
values of the function u to be determined for the whole area, and will be 
called the assigned values. Let the maximum of the values be M and the 
minimum be m ; and denote M m by /*, so that p is positive. 

Assume, for a boundary value along L 2 , the minimum m of the assigned 
values for the function along L and L 3 . Let the function, which is uniquely 
determined for the region 2\ by the general conditions for the area and by 
values along the boundary, constituted by the assigned values along L and 
the assumed value m along L 2 , be denoted by i^. The values assumed by u^ 
along the line L in this region are uniform, finite and continuous ; and they 
may be denoted by m+p/j,, where p is a positive proper fraction varying from 
point to point along the line. 

Let the function, which is uniquely determined for the region T 2 by the 
general conditions for the area and by values along the boundary, constituted 
by the assigned values along L 3 and by the values of Wj along L 1} be denoted 
by u z . Then the uniform, finite, continuous values which it assumes along 
L 2 are of the form m + qp, where q is a positive proper fraction varying from 
point to point along the line ; let the greatest of these values be m + Q/J,, 
where Q lies between and 1. 

For the region T determine a function* u 3 by means of boundary values, 
consisting of the assigned values along L and the values of u 2 , viz., m + Q/J,, 
along -Z/j. Then the function u 3 u-^ satisfies the general conditions ; its 
value along the part L of the boundary is zero, and its value along the 
other part L 2 of the boundary is < Q/JL and is greater than zero. Hence u 3 u 
is always positive within T , and along L^ we have u 3 u^ ^ qiQ/J*. 

For the region T 2 determine a function M 4 by means of boundary values, 
consisting of the assigned values along L 3 and the values of u 3 along L^ 
Then the function u 4 u 2 satisfies the general conditions; its value is zero 
along Z 3 ; and its value along L^ is that of u 3 u lt that is, a positive quantity 
which is not greater than qfyfi. Hence w 4 u 2 is always positive within T 2 , 
and along L. 2 we have u 4 - u 2 



* All the succeeding functions will be determined subject to the general conditions for 
the respective areas ; the specific mention of the general conditions will be omitted. 



222.] OF AREAS 427 

For the region T^ determine a function u 5 by means of boundary values, 
consisting of the assigned values along L and the values of w 4 along L 2 . 
Then the function u 5 u s satisfies the general conditions ; its value is zero 
along L \ and its value along L 2 is that of w 4 w 2 , that is, a positive quantity 
which is not greater than q 2 qiQp. Hence u 5 u 3 is always positive within T^ , 
and along L 1 we have u s u 3 ^ (ffli QfL 

Proceeding in this manner for the regions alternately, we obtain functions 
UMI+I for the region T 1} such that u m+l has the assigned values along L and 
the values of u. M along L 2 ; and functions u, 2n for the region T 2 , such that u^ n 
has the assigned values along L 3 and the values of u m -i along L^. And the 
functions are such that 

Man-i > in ^ and ^ q^q^ 1 Qp along L ; and 
u-2n > in T 2 and ^ q^q. Qp along L 2 . 

Hence, both for functions with an uneven suffix and for functions with an 
even suffix, there are limits to which the functions approach along L and L. 2 
respectively ; let these limits be u and u". 

Both of these limits are finite ; for along L 1} we have 

u = HI + (u 3 - MJ) + (u, u 3 ) + . . . ad inf. 



so that this expression, which is finite, is an upper limit and m is a lower 
limit for u . And, along L 2 , we have 

u" = u 2 + (w 4 M 2 ) + (u 6 w 4 ) -f ... ad inf. 
<; m + Q/J, + qiq 2 Q/j, + q^q 2 2 Q/J, + ... 

^ ml Qn 

so that this expression, which is finite, is an upper limit and m is a lower 
limit for u". Hence both u and u" are finite. 

Now in determining u for T l and regarding it as the limit of u 2n+1 , we 
have its values along L 2 as the values of w 2n , that is, of u" in the limit ; and 
in determining u" for T 2 and regarding it as the limit of u 2n+2 , we have its 
values along D, as the values of u 2n+1 , that is, of u in the limit. Hence over 
the whole boundary of T, the region common to T l and T a , we have u = u" ; 
and therefore (by 220, VI) we have u =u" over the whole area of the 
common region T. 

Lastly, let a function u be determined for the region T 1} having the 
assigned values along L and the values of u along L. 2 . Then the function 
u - u satisfies the general conditions ; it has zero values round the whole 



428 COMBINATION OF AREAS [222. 

boundary of T l , and therefore (by 220, VI) it is zero over the whole region 
T!. Hence u is the function for T l . 

Similarly, determining a function u for T 2 , having the assigned values 
along L s and the values of u" along L lf we have u = u" everywhere in T 2 , so 
that u" is the function for T^. 

The functions u and u" satisfy the general conditions for T l and T 2 
respectively ; and these two regions have a common portion T over which 
u! and u" have been proved to be the same. Hence, by Lemma I. of 216, 
they determine one and the same function for the whole region combined of 
T l and T 2 ; this function u satisfies the general conditions and, along the 
boundary of the whole region, assumes values that are assigned arbitrarily 
subject only to the general limitations of being everywhere finite and, 
except for finite discontinuities at isolated points, uniform and continuous. 
The proposition is therefore established. 

This method of combination, dependent upon the alternating process 
whereby a function determined separately for two given regions having a 
common part is determined for the combination of the regions, is capable of 
repeated application. Hence it follows that a function exists, subject to the 
general conditions within a given region and acquiring assigned finite values 
along the boundary of the region, when the region can be obtained by 
combinations of areas that can be conformally represented upon the area of a 
circle. 

Note. Let A, B, C be three non- intersecting simple closed curves, such 
that G lies within B and B within A. The area bounded by the curves A and 
C can, by a similar method, be combined with the whole area enclosed by B ; 
and we can make the same inference as above, as to the existence of a function 
u for the whole area enclosed by A, when it exists for the areas that are 
combined. 

223. At the beginning of the discussion it was assumed that the areas, 
in which the existence of the function is to be proved, lie in a single sheet 
( 216) or, in other words, that no branch-point occurs within the area. 

It is now necessary to take the alternative possibility into consideration : 
a simple example will shew that the theorem just proved is valid for an area 
containing a branch-point except in one unessential particular. 

Let the area be a winding surface consisting of m sheets : the region in 
each sheet will be taken circular in form, and the centre c of the circles will 
be the winding-point, of order m I. Such a surface is simply connected 
( 178) ; and its boundary consists of the m successive circumferences which, 
owing to the connection, form a single simple closed curve. Using the 
substitution 

z - c = RZ M , 



223.] BRANCH-POINT IN AREA 429 

we have a new ^-surface which consists of a circle, centre the Z-origin and 
radius unity : it lies in one sheet in the ^-region and has no branch-points ; 
its circumference is described once for a single description of the complete 
boundary of the winding-surface. The correspondence between the two 
regions is point-to-point: and therefore the assigned values along the bound 
ary of the winding-surface lead to assigned values along the ^-circumference. 
Any function w of z changes into a function W of Z: hence u changes 
into a real function U satisfying the general conditions in the ^-region ; 
and conversely. 

But a function U, satisfying the general conditions over the area of a 
plane circle and acquiring assigned finite values along the circumference, is 
uniquely determinate ; and hence the function u is uniquely determined on 
the circular winding-surface by satisfying the general conditions over the area 
and by assuming assigned values along its boundary. 

It is thus obvious that the multiplicity of sheets, connected through 
branch-lines terminated at branch-points and (where necessary) on the single 
boundary of the surface consisting of the sheets, does not affect the validity 
of the result obtained earlier for the simpler one-sheeted area ; and therefore 
the function u, acquiring assigned values along the boundary of the simply 
connected surface and satisfying the general conditions throughout the area 
of the surface which may consist of more than a single sheet is uniquely deter- 
\ minate. 

There is, as already remarked, one unessential particular in which 
deviation from the theorem occurs when the region contains a branch-point. 
At a branch-point a function may be finite*, but all its derivatives are not 
necessarily finite ; and therefore at such a point a possible exception to the 
general conditions arises as to the finiteness of value of the derivatives 
and the consequent satisfying of the equation V-u = : no exception, of 
course, arises as regards the uniformity of the derivatives on the Riemann s 
surface. The exception does not necessarily occur ; but, when it does occur, 
it is only at isolated points, and its nature does not interfere with the validity 
of the proposition. We shall therefore assume that, in speaking of the 
general conditions through the area, the exception (if necessary) from the 
general conditions, of finiteness of value of the derivatives at a branch-point, is 
tacitly implied. 

Hence we infer, by taking combinations of circles in a manner some 
what similar to the process adopted for successive circles of convergence 
in the continuation of a function in 34, that a function u exists, subject to the 
general conditions within any simply connected surface and acquiring assigned 
finite values along the boundary of the surface. 

* Infinities of the function itself at a branch-point will fall under the general head of infinities 
of the function, diecnssed afterwards (in 229). 



430 



MODULI OF PERIODICITY 



[224. 



224. The functions which have been discussed so far in the present 
connection are functions which have no infinities and, except possibly at 
points on the boundaries of the regions considered, no discontinuities : they 
are uniform functions. And the regions have, hitherto, been supposed simply 
connected parts of a Riemann s surface, or simply connected surfaces. When 
the surface is multiply connected, we resolve it by a canonical system 
( 181) of cross-cuts and proceed as follows. 

We now proceed to introduce the cross-cut constants, and so to consider 
the existence of functions which have the multiform character of the integrals 
of uniform functions of position on the Riemann s surface. The functions 
will still be considered to be uniform, finite and continuous except at the 
cross-cuts : their derivatives will be supposed uniform, finite, and continuous 
everywhere in the region, and subject to the equation V 2 u = : and boundary 
values will be assigned of the same character as in the previous cases. As 
moduli of periodicity are to be introduced, the unresolved surface is no longer 
one of simple connection : we shall begin with a doubly connected surface. 

Let such a surface T be resolved, in two different ways, into a simply 
connected surface : say into T t by a cross-cut Q x , and into r l\ by a cross 
cut Q 2 . Mark on T^ and on T 2 the directions of Q 2 and of Q 1 respectively : the 






Fig. 81. 

notations of the boundaries are indicated in the figures, and T is the 
region between the lines of Qj and Q 2 . 

It will be shewn that a function u exists, determined uniquely by the 
following conditions : 

(i) The first and the second derivatives are throughout T to be 
uniform, finite and continuous, and to satisfy V"u = : but no conditions 
for them are assigned at points on the boundary : 

(ii) The (single) modulus of periodicity is to be K, which will be 
taken as an arbitrary, real, positive constant : the value of any branch of u at 
a point on the positive edge is therefore to be greater by K than its value at 
the opposite point on the negative edge : 

(iii) Some selected branch of u is to assume assigned values along 



224.] FOR MULTIPLY CONNECTED SURFACES 431 

a and b , typically represented by H, and assigned values along a and b, 
typically represented by G. These boundary values are to be finite every 
where, though they may be discontinuous at a finite number of isolated points 
on the boundary ; such discontinuity will arise through the modulus. 

In T 1} for zero values along a, b, a , b and for unit values along Q{~ 
and Qj + , let the fractional factor for the line Q. 2 be q 1 : and similarly in T 2 , 
for zero values along a, b, a , b and for unit values along Q 2 ~ and Q 2 + , 
let the fractional factor for the line Qi be q 2 , where <ft and g 2 are positive 
proper fractions. 

For the simply connected region* T-i determine a function u 1} satisfying the 
general conditions and having as its boundary values, H along a and b , G 
along a and b, arbitrarily assumed values represented by 6 (the maximum 
value being Mj. and the minimum value being m^ along Q~ and values 
B + K along Qi + : the function so obtained is unique. Let the values 
along the line Q 2 in ^i be denoted by w/. 

For the region T 2 determine a function w 2 , satisfying the general 
conditions and having as its boundary values, H along a and b , G K 
along a and 6, u^ K along Q 2 ~ and u-[ along Q 2 + : the function so ob 
tained is unique. Let its values along the line Q l in T a be denoted by 
u 2 , the maximum value being M t and the minimum value being m 2 . 

For the region T l determine a function u s , satisfying the general conditions 
and having as its boundary values, H along a and b , G along a and b, u 2 
along Qr and u. 2 + K along Qj+ : the function so obtained is unique. Let its 
values along the line Q 2 in T t be denoted by u 3 . Then the function u s u : 
satisfies the general conditions in T l ; it is zero along a and b , a and b : it is 
u, f - along Qj- and also alo